There are a few words you should know the meaning of::::::
Definitionsssssssss
Click on the blue words
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Page 3 of 12
suffix
suffix
In order to distinguish matrix elements from each other, SUFFIX NOTATION is used to refer to a particular element.
For example, in the matrix A , the element common to the 2nd row and 4th column is referred to as a24.
So in this example a24 = -6...atrix are referred to by lower-case letters.
2 7 8 1
-1 1 -1 3
7 4 -1 2
3 -2 4 -6
-1 1 -1 3
2 7 8 1
-1 1 -1 3
leaddiag
2 7 8 1
-1 1 -1 3
7 4 -1
3 -2 4
-1 1 -1
2 7 8 1
-1 1 -1 3
M = 4 -1
3 -2 4
-1 1 -1
2 7 8 1
-1 1 -1 3
leaddiag
In an n x n square matrix M, the elements m11 , m22 , m33 , ..... , mnn are referred to as the LEADING DIAGONAL of M.
In this example, the leading diagonal (7, -2, -1) is coloured green.erred to as the LEADING DIAGONAL of M............
2 7 8 1
-1 1 -1 3
7 4 -1 2
3 -2 4 -6
-1 1 -1 3
2 7 8 1
-1 1 -1 3
element
Each number in the matrix is referred to as an ELEMENT.
In this example there are 12 elements:
order
A matrix with m rows and n columns is referred to as an m x n matrix: such a matrix is said to be of ORDER m x n.
In this example the matrix is of order
3 x 4:
REMEMBER: order is ROWS x COLUMNS....
square
If a matrix has n rows and m columns and m = n, the matrix is called a SQUARE matrix.
In this example the matrix is a 3 x 3 square matrix:::::
scalar
If a matrix has just one row and just one column, then it only has a single element. The matrix is then called a SCALAR.
A scalar is usually written without the brackets, eg.
7. 7.
2 7 8 1
-1 1 -1 3
7 4 -1
3 -2 4
-1 1 -1
2 7 8 1
-1 1 -1 3
starttext
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ELEMENTS
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ORDER
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SQUARE MATRIX
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SCALAR
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SUFFIX NOTATION
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LEADING DIAGONAL
wwwwwwww
wwwwwwww
wwwwwwww
wwwwwwww
wwwwwwww
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wwwwwwww
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identity
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identity
Two matrices that deserve a special mention are:
(1)
the 3x3 identity matrix. The identity matrix (of any order) is denoted by I .
ts are 0. Thus
is the 3x3 identity matrix. The identity matrix (of any order) is denoted by I .
er) is denoted by I .
is the 3x3 identity matrix. The identity matrix (of any order) is denoted by I .
ny order) is denoted by I .
x. The identity matrix (of any order) is denoted by I .
is the 3x3 identity matrix. The identity matrix (of any order) is denoted by I .
0 0 ]
[ 0 1 0 ]
[ 0 0 1 ]
is the 3x3 identity matrix. The identity matrix (of any order) is denoted by I .
matrix. The identity matrix (of any order) is denoted by I .
ultiplying another matrix by I gives that matrix.
Definitions
Click on the red matrices
2 of 4
Page 4 of 12{
The IDENTITY matrix.
Click the matrix for more information.
identity
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identity
The ZERO matrix.
Click the matrix for more information.
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1 0 0
0 1 0
0 0 1
2 7 8 1
-1 1 -1 3
2 7 8 1
-1 1 -1 3
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0 0
0 0
0 00000
2 7 8 1
-1 1 -1 3
2 7 8 1
-1 1 -1 3
The ZERO, or NULL, matrix of order mxn, is an m by n matrix ALL of whose entries are ZERO.
Multiplying a matrix by a zero matrix gives a zero matrix (compare this with the number zero).
identity
The IDENTITY matrix is a square matrix in which elements on the leading diagonal are all ONE and ALL other elements are ZERO.
Multiplying a matrix by an identity matrix gives the matrix you started with (compare this with the number one).
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Summary
applications
matex2
column
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In an mxn matrix, if m = 1 or n = 1 (but not both) then the matrix is commonly referred to as a VECTOR, eg.
(1)
(2) ( 6 4 -1 )
( 2 0 -1 -6 )
You can find out more about this in the Vectors module...
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Definitions
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Page 5 of 12I
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2 7 8 1
-1 1 -1 3
-2 000000000000000
2 7 8 1
-1 1 -1 3
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column
2 7 8 1
-1 1 -1 3
-222222 000000000000000
2 7 8 1
-1 1 -1 3
If a matrix has only one row (m = 1), we refer to it as a ROW VECTOR.
So these are examples of 1x3 and 1x4 row vectors.s. vectors.
column
If a matrix has only one column (n = 1), we refer to it as a COLUMN VECTOR.
So these are examples of 3x1 and 2x1 column vectors.
Click on the red vectors
Definitions
Introduction
matex3
Summary
You should now understand the following about matrices and their properties:
Definitions of terminology such as elements, order, square matrix,
leading diagonal and so on;
Symmetry and how to transpose a matrix;
Equality of matrices;
How and when one can add and subtract matrices;
How and when one can multiply matrices.....
Summary
1 of 1
Page 12 of 12
matex1
matrix algebra
scalar
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scalar
#69699
Equality of matrices
Two mxn matrices A and B can be equal only if all of their elements are equal.
Multiplication by a scalar
If k is a scalar and A is a matrix then the PRODUCT, kA, is given by multiplying each element of A by k.
scalar
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Matrix Algebra
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showme
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Show me an example of SCALAR MULTIPLICATION
scalartext
4 0
4 0
10 5 4 0
2 1
3 55
0
4 0
10 5 4 0
3*2 3*1
3*3 3*55
4 0
4 0
10 5 4 0
6 3
9 15x5555
This principle of SCALAR MULTIPLICATION
holds for matrices of ANY order.....
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Show me an example of MATRIX EQUALITY
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, B =
and A = B
equalitytext
then we must have
a = -6.22 = 4 = 2 b21 = 3 b22 = 4
4 0
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a 12
3 444444
4 0
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3 422
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A scalar is a single number (i.e. a matrix that has only one column and one row), e.g. 5 or 12.4.
Click to removeeeeeeeee
If two matrices A and B can be multiplied to get C (= AB), then the elements of C are formed by multiplying the ROWS of A by the COLUMNS of B.
fact, the elements Cij are formed by multiplying the i'th row of A by the j'th column of B.
on of how the element
c is formed from the above two matrices A and B.
From the above formula, we must take the "product" of row 2 of A and
column 1 of B. * ]
C = [ * * ]
[ * * ]
What are C's elements? As stated above, we mutiply the ROWS of A with the
COLUMNS of B and add the results. Specifically,
c = sum of products (ROW i of A) and (COLUMN j of B)
ij of elements of
As an example, we'll now show you an animation of how the element
c is formed from the above two matrices A and B.
From the above formula, we must take the "product" of row 2 of A and
column 1 of B. [ 7 0 ]
A = [ 8 0 5 3 ] B = [ 1 8 ] .
[ 1 2 1 2 ] [ 2 2 ]
[ 4 0 ]
We know, from the multiplication rule, that A and B can indeed be multiplied,
and that their product -- we'll call it C -- must be a 3x2 matrix. So C looks like
[ * * ]
C = [ * * ]
[ * * ]
What are C's elements? As stated above, we mutiply the ROWS of A with the
COLUMNS of B and add the results. Specifically,
c = sum of products (ROW i of A) and (COLUMN j of B)
ij of elements of
As an example, we'll now show you an animation of how the element
c is formed from the above two matrices A and B.
From the above formula, we must take the "product" of row 2 of A and
column 1 of B. trix. So C looks like
[ * * ]
C = [ * * ]
[ * * ]
What are C's elements? As stated above, we mutiply the ROWS of A with the
COLUMNS of B and add the results. Specifically,
c = sum of products (ROW i of A) and (COLUMN j of B)
ij of elements of
As an example, we'll now show you an animation of how the element
c is formed from the above two matrices A and B.
From the above formula, we must take the "product" of row 2 of A and
column 1 of B.
Matrix Algebra
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Page 10 of 12
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Click on the buttons opposite for information on matrix multiplication......
theory
theory
In fact, a single element cij, of the product of the matrices A and B, is formed by multiplying the ith row of A by the jth column of B.
This multiplication of a row by a column is carried out by multiplying each element of the row by the corresponding element of the column, then all these products are added together. This process is known as the vector scalar product.tion.st element of the column and to this is added the product of the second elements of the row and column etc..
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If the matrices A (order nxm) and B (order mxp) are multiplied to form C (order nxp) then the element c is formed by multiplying row 1 of A by column 2 of B, i.e
c = ( a a a ..... a )
= a b + a b + .... + a b
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Consider the matrix product
AxB = x =
lick onto the small buttons to
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or the animate button for the
of other orders.
s for matrices of other orders.
N*f)K*
2 7 8 1
-1 1 -1 3
2 7 8 1
-1 1 -1 3
$.<-!.
2 7 8 1
-1 1 -1 3
5+3 7+6
4+1 1+0
-2+6 3+233
2 7 8 1
-1 1 -1 3
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Click onto the small buttons to see individual elements formed, or the animation button for the complete matrix.
4 1 0
4 1 0
2 7 8 1
-1 1 -1 3
2 2222222222233
2 7 8 1
-1 1 -1 3
3 2 5
4 1 0
2 7 8 1
-1 1 -1 3
1 22222222233
2 7 8 1
-1 1 -1 3
times
scalarproduct
The row and column are vectors, each with the same number of elements. The scalar product of two vectors is only defined when both vectors have the same number of elements. It is this fact that gives rise to the matrix multiplicative condition (no. columns of A = no. rows of B).
See the Vector Module for a detailed explanation of scalar product. t.
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matex1
5~687
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There are two further properties of matrices which you should know about.
Click onto the exercise book to test your understanding of these properties...
Definitions
4 of 4
Page 6 of 12A
answer(2)
answer(1)
TRANSPOSE AND SYMMETRY
Exercise 111e 1D SYMMETRY
Exercise 1
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The transpose of a matrix
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Symmetric matrices
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Matrix A is symmetric if and only if A = AT. Hence note that only square matrices can be symmetric.
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Let A be an m by n matrix. The transpose of A, denoted by AT, is the matrix which is formed by interchanging the rows and columns of A.
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transpose
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2 3 -1
6 -4 3
-7 2 2
AT =
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P0(0M0
2 -1
-6 3
-7 2
B =
X101U1
BT =
2 6 -7
6 -4 2
-7 2 2
s2brackets
AT =
2 6 -7
3 -4 2
-1 2 2
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T6,6Q6
BT =
a2c1r1
2 6 -777 2
2 2
a2c1r2
3 -4 222 2
2 2
a2c1r3
-1 3 2222 2
2 2
a2c2r1
6 -7 6 -777 2
2 2
a2c3r1
-7 6 -777 2
2 2
a2c2r2
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2 2
a2c3r2
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2 3 -4 222 2
2 2
a2c2r3
3 2 3 2222 2
2 2
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2-1 3 2222 2
2 2
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2 -6 -777 2
2 2
a4c1r2
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2 2
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-6 -7-6 -777 2
2 2
a4c3r1
-7 -6 -777 2
2 2
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3 -1 3 222 2
2 2
a4c3r2
2 -1 3 222 2
2 2
2 6 -777 2
2 2
6 -4 222 2
2 2
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2 2
s4c3r1
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2 2
s4c1r2
6 -4 22 2
2 2
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2 2
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-7 2 2
2 2
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2 2
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2 6 -717 2
2 2
s4c2r1
3 -4 222 2
2 2
exback
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A =
C = A + B
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Exercise 111111
Symmetry and Transposeeeeeeeeeeeen
question
Question 181111
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Is A symmetric ? Click the appropriate buttonnr
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Use the Tab key (or mouse) to move from element to element. Enter the appropriate value for each element. Once all the elements have been entered click the Enter key...
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What is the transpose of A, i.e AT ?
AT =???????????????????????????????
ansback
ansback
The transpose is formed by swapping the rows
and columns of the matrix . Since
A = then AT =
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a3c1r1
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2 2
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s1text
A = AT, therefore A is symmetric.
matrices can be symmetric.
Click on the numbered buttons to see some examples..........................................d buttons to see some examples.
a1c1r1
-7 2 2
2 2
a1c2r1
ts,sqs
2 2 2
2 2
a1c3r1
2 2 2
2 2
a1c1r2
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2 2
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2 2 2
2 2
s3c1r1
2 6 -717 2
2 2
s3c2r1
s3c3r1
2 2 2
2 2
s3c1r2
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2 2
s3c2r2
2 2 2
2 2
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2 2
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2 2 2
2 2
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2 2
s2text
BT, therefore B is NOT symmetric.
rices can be symmetric.
Click on the numbered buttons to see some examples..........................................d buttons to see some examples.
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i&"r3")
i&"r1")
i&"r2")
i&"r3")
movef3 ("
i&"r1"),("a4c1r"&i),("
i&"r2"),("a4c2r"&i),("
i&"r3"),("a4c3r"&i),35
i&"r1")
i&"r2")
i&"r3")
i&"r1")
i&"r2")
i&"r3")
clearup 3
animate
animateold
s2brackets
("s1c"&j&"r"&i)
flashf ("
fpos
movef ("
M,("s2c"&i&"r"&j)
"s1text"
Zpoint fpos1,fpos2,fpos3
("s1c1r"&i)
("s1c2r"&i)
("s1c3r"&i)
sbmflashfff ("
movef3 ("
1"),("
2"),("
3"),35
clearup 4
animate
animateold
s4brackets
("s3c"&j&"r"&i)
flashf ("
fpos
movef ("
M,("s4c"&i&"r"&j)
"s2text"
Zpoint fpos1,fpos2,fpos3
("s3c1r"&i)
("s3c2r"&i)
("s3c3r"&i)
sbmflashfff ("
movef3 ("
1"),("
2"),("
3"),35
onum
"inner"
"textouter"
order
"pageno"
"page3"
K= "Again?"
K= "More"
"multcondition"
180,50,100
180,50,100
showex exname
4prevbutton
starttext
element
hideobj
order
square
scalar
suffix
leaddiag
4matmultflag
isinterrupt
B"c11"
B"c12"
B"c21"
B"c22"
B"animate"
"Again ?"
,oldmatmultflag
a,b,c,d,x,y
=(1)
enabled
"Continue ?"
logorder
4corder,numberRight,FirstTry
tbkscore[][]
firstTry =
enabled
B"enter" =
"correct"
X1][1]=
"question"
v1][2]=numberright
showinputmatrix
-- code found
"tabtext"
"Incorrect, answer should be "&(
f"Next Question"
"Quit Exercise"
B"QuitEx"
tidyup
genquestion
"ex2"
4animateflag
garraypos[6]
> 0)
( < 6)
isinterrupt
sumelement
B"addanimate"
--
"Again ?"
,oldanimateflag
a,b,c,d,x,y
=(1)
"Continue ?"
"order2"
startmatrix
pluscol
plusrow
exback
Bquitex
4numberRight
tbkscore[][]
2][1]=0
) = 0
"question" = 0
B" = 0
V" = 0
m2][2]
2][1]
2][1]
exinuse
expage = "mat"&(
8= "ex2"
B"QuitEx"
"matex2"
"exback"
"tabexplan"
genquestion
4acount,mcount,corder,firsttry
enabled
B"enter" =
(3) = 3
m1n =
(2)+1
m1m =
(2)+1
m2n =
(2)+1
m2m =
(2)+1
(2)+1
(2)+1
genmatrix
q,-2,9,2018,2915
,-2,9,2018,2780
,-2,9,4208,2915
,-2,9,4208,2780
"mg1"
"mg2"
"order2"
exists
$&"x"&
4row,col,biggest[10]
"matrix"
-- ignore
input phase
exercise
U>47
]<58)
f=43
-- only handles integer values,+,-
" > 9000
"You shouldn't need elements
width"
= "?"
crow =
current
ccol =
resizeElement
7,120
-- BackSpace
caretlocation
equalbiggest(
F-- no other
same
,-120
-- Tab
--
enabled
B"enter" =
-- rightarrow
-- leftarrow
4row,col,biggest[10]
"matrix"
-- ignore
input phase
exercise
U>47
]<58)
f=43
-- only handles integer values,+,-
" > 9000
"You shouldn't need elements
width"
crow =
current
ccol =
resizeElement
7,120
-- BackSpace
caretlocation
equalbiggest(
F-- no other
same
,-120
-- Tab
--
enabled
B"enter" =
-- rightarrow
-- leftarrow
4but1A,but2A, but3A,but4A,numberRight,attempted
tbkscore[][]
numberWrong=0
"smile1"
"frown1"
"smile2"
"frown2"
"smile3"
"frown3"
"smile4"
"frown4"
= 0
3][1]=
question
3][2]=numberright
"Correct"
f"Next
"Quit exercise"
3][1]=
= 1
"Incorrect:
sad face indicates
incorrect choice"
B"QuitEx"
tidyup
genquestion
"ex3"
4row,col,biggest[10]
"matrix"
-- ignore
input phase
exercise
U>47
]<58)
f=43
n=45
v = 42
-- only handles integer values,+,-,*
" > 9000
"You shouldn't need elements
width"
crow =
current
ccol =
resizeElement
7,120
-- BackSpace
caretlocation
equalbiggest(
F-- no other
same
,-120
-- Tab
--
enabled
B"enter" =
-- rightarrow
-- leftarrow
4row,col,numberRight
tbkscore[][]
4logical firstTry
\i,j,numberWrong,index,index2
stext
) = 0
checkfieldsyntax ("mat"&i&j)
studans =
g(i+(j-1)*
"m1" )
= 0
= 0
numberright
1][2]=
"Correct"
f"Next question"
"Quit exercise"
= 1
"Incorrect:
the highlighted value indicates
incorrect element"
f"Try Again"
"Show Answer"
^lues
1][1]
tidyup
B"QuitEx"
"matrix"
genquestion
"ex1"
oenabled =
"ansback"
"mg1"
H-100,300
row = 2
-- generate
called "mg2"
genmatrix
-4,-2,4900,3215
-4,-2,4900,3080
= (j-1)*
wtext =
= " "&
"m2" =
4row,col,numberRight
tbkscore[][]
4logical firstTry
\i,j,numberWrong,index
msg,mtext
& = 0
checkfieldsyntax ("mat"&i&j)
studans =
"m1" +
"m2")
= 0
= 0
2][1]=
"question"
2][2]=numberright
firsttry=
"Correct"
f"Next
"Quit exercise"
2][1]=
= 1
"Incorrect:
the highlighted value indicates
incorrect element"
f"Try Again"
"Show Answer"
^lues
tidyup
B"QuitEx"
"matrix"
genquestion
"ex2"
oenabled =
"ansback"
"mg1"
H-100,0
"mg2"
H-760,0
row = 2
-- generate
called "mg3"
genmatrix
0,3,4900,2915
0,3,4900,2780
"m3" =
width = (
)/2 + 700
!"a11"
!"a12"
f+100)
"m3" =
col-1
= (i-1)*
"m2" = "-"
"+"&(
"m2")
0,3,7615,2915
0,3,7615,2780
"mg4" =
!"a15"
!"a16"
+100)
"m4" =
tmpchar = (
"m1")+(
"m2")
3) = 1
"m4" = (
"m4" = (
{&" ")
4numberRight
tbkscore[][]
1][1]=0
) = 0
"question" = 0
B" = 0
V" = 0
m1][2]
1][1]
1][1]
exinuse
expage = "mat"&(
8= "ex1"
B"QuitEx"
"matex1"
"exback"
"tabexplan"
genquestion
4acount,mcount,corder,FirstTry
m1n,m1m,rannum
ltext
enabled
B"enter" =
firstTry =
(2) = 1
u = 2
z = 2
= 2
genmatrix
-8,9,2018,2915
-- now alter elements i.e.
-- want some matrices symmetric, others nearly so
= "2x2 nonsymm"
--
= changeval(2,3,
= "2x2
-- row 1 =
asure
= sbmrandom2(-9,-1,
,0,9,
(3) = 3
aterms on leading diag equal
-8,9,2018,2780
-- 3x3
--
i.e.
--
= "3x3
--
= "3x3
--
(3) = 3
"m1" =
"mg1"
"order2"
m1n=m2n
m1m=m2m
exists
(&"x"&
ochangeval oldword,newword,stext
oldval,newval, remchar
5 < 0
7 < 0 )
J >= 0
M >= 0)
t < 0
= " "&
logorder
4corder,numberright,firstTry
tbkscore[][]
"correct"
04][2]=
D" = numberRight
enabled
B"enter" =
showinputmatrix
-- code found
"tabtext"
-- may have been hidden
Ha large
Jquestion
"Incorrect, answer should be "&(
f"Next Question"
"Quit Exercise"
B"QuitEx"
tidyup
genquestion
"ex4"
4][1]=
FirstTry
"order2"
startmatrix
pluscol
plusrow
showexplan buttonname
V,m1m
crow,ccol
\" = "C"&
@&" = "
x-1)*
"m1"
g(k-1)*
"m2"
" = "
exback
Bquitex
logorder
4corder,numberRight
tbkscore[][]
1][1]
enabled
B"enter" =
"Correct"
m1][2]=numberright
= "nonsymm"
showinputmatrix
-- code found
"tabtext"
tidyup
B"QuitEx"
genquestion
"ex1"
"Incorrect, answer should be Yes"
f"Next Question"
y Exercise"
flashrr g,h
clearup n
("a1c"&j&"r"&i)
("a2c"&i&"r"&j)
"a2eq"
"a2brackets"
("a3c"&j&"r"&i)
("a4c"&i&"r"&j)
"a4eq"
"a4brackets"
("s1c"&j&"r"&i)
("s2c"&i&"r"&j)
"s2eq"
"s2brackets"
"s1text"
("s3c"&j&"r"&i)
("s4c"&i&"r"&j)
"s4eq"
"s4brackets"
"s2text"
firstTry
"order2"
startmatrix
pluscol
plusrow
exback
Bquitex
4numberRight,firstTry
tbkscore[][]
total
"question")-1
L1][1]=
X1][2]=numberright
exinuse
"ex1"
exer=1
][1]>0
performance(
%][2],
.][1])
storemarks bkid
T][1],
]][2]
i][1]=0
w][2]=0
"exback"
"ansback"
tidyup
"order2"
"matrix"
sbmnavigate
4acount,mcount
("mg"&i)
clearmatrix
-- now
4numberRight,firstTry
tbkscore[][]
total
"question")-1
L2][1]=
X2][2]=numberright
exinuse
"ex2"
exer=2
][1]>0
performance(
%][2],
.][1])
storemarks bkid
T][1],
]][2]
i][1]=0
w][2]=0
"exback"
tidyup
"order2"
"matrix"
sbmnavigate
4acount,mcount
("mg"&i)
clearmatrix
-- now
4numberRight,attempted
tbkscore[][]
total =
"question"
M3][1]=
Y3][2]=numberright
exinuse
8= "ex3"
exer=3
][1]>0
performance(
%][2],
.][1])
storemarks bkid
T][1],
]][2]
i][1]=0
w][2]=0
"exback"
"blayer"
tidyup
sbmnavigate
status="Click
board
exercise"
4acount,mcount
("mg"&i)
"smile1",
"smile2",
"smile3",
"smile4",
"frown1",
"frown2",
"frown3",
"frown4"
B"but1" =
B"but2" =
B"but3" =
B"but4" =
4numberRight
tbkscore[][]
status=""
3][1]=0
8 = 0
"question" = 1
Q" = 0
e" = 0
|3][2]
3][1]+1
3][1]
exinuse
expage = "mat"&(
8= "ex3"
B"QuitEx"
"matex3"
"exback"
"blayer"
genquestion
4acount,mcount,but1A,but2A, but3A,but4A,attempted
m1n = 1
m1m =
(3)+1
m2n =
(3)+1
m2m = 1
m3n =
(3)+1
m3m =
(3)+1
m4n =
(3)+1
m4m =
(3)+1
(3)+1
(3)+1
choice =
m0m =
m0n =
(3)+1
genmatrix
,m1m,-2,7,1760,4895
,m2m,-2,7,3830,4895
,m3m,-4,7,6000,4895
{,m4m,-2,7,8070,4895
m0n,m0m,-2,9,5300,2575
-- the
"mg5"
)/2 - 200
"b" = (
+ 150),
bringtoFront
4numberright
tbkscore[][]
status=""
4][1]=0
numberRight = 0
"question" = 1
Z" = 0
n" = 0
4][2]
4][1]+1
4][1]
exinuse
expage = "mat"&(
8= "ex4"
B"QuitEx"
"matex4"
"exback"
"tabexplan"
genquestion
4acount,mcount,corder,m1m,FirstTry
firstTry=
enabled
B"enter" =
ran =
m1n = 2
b = 2
(3) = 3
m2n = 3
m2m = 2
(3) = 3
m1m = 2
(3) = 3
(3) = 3
genmatrix
-2,6,2018,2390
-2,5,2018,2255
m2n,m2m,-2,6,4208,2390
5,4208,2255
"mg1"
"mg2"
"order2"
-- product exists
&"x"&
4numberright,firstTry
tbkscore[][]
total=(
"question")-1
L4][1]=
X4][2]=
exinuse
8= "ex4"
exer=4
][1]>0
performance(
%][2],
.][1])
storemarks bkid
T][1],
]][2]
i][1]=0
w][2]=0
"exback"
tidyup
"order2"
"matrix"
status="Click on
exercise
sbmnavigate
4acount,mcount
("mg"&i)
'*4+1)
!("a"&i)
clearmatrix
-- now
4animateflag,oldanimateflag
"Again ?"
c11f,
c12f,
c21f,
c22f,
c31f,
i = 1
B"interrupt"
"Continue ?"
enabled
"animation"
.tbk"
questionmark
-- see
handler
sumelement pos
temp1
("a"&
temp2
"plus"
temp3
("b"&
("a"&
("b"&
sbmflashfff ("a"&
u),("b"&
~), "
movef3 ("a"&
),pos1,
v,pos2,("b"&
),pos3
-- moves aij, plussign
positions above the matrices
-- these are later replaced
Ha single
which
Fmoved
Danswer matrix
("c"&
("a"&
("b"&
("a"&
("b"&
("c"&
), ("c"&
&"f")
("c"&pos)
("c"&
&"f")
("c"&
-- handlers
simon.tbk
4matmultflag,oldmatmultflag
"Again ?"
@ = 1
B"interrupt"
"Continue ?"
enabled
"animation"
.tbk"
-- see
handler
sumelement row,col,nelements
-- routine which takes a
a column (
2 matrices
animates their product.
flashff ("
posa =
"pos1"
posb =
"pos2"
posc =
"pos3"
diff = (
temp1
("a"&row&i)
temp2
"times"
temp3
("b"&i&col)
("a"&
("b"&i&
sbmflashfff ("a"&
l,("b"&i&
movef3 ("a"&
,("b"&i&
("c"&
("a"&
("b"&i&
("a"&
("b"&i&
("a"&
("b"&i&
("c"&
("c"&
", ("c"&
B("c"&
("c"&
("row"&
("col"&
-- handlers
simon.tbk
flashwe word1,word2,ellipse1,tfield, colour
4pauseanime, mytime
\iter,i
Zlogical fxbreak
= 4 +
-/400
ncolour
sycpause 10
adjustspeed()
reset
flasharc arcname, numsegments,
= 4 +
flashline linename,
= 4 +
oflashfield fieldname
= 4 +
4row,col,m1m,numberright,firstTry
tbkscore[][]
msg,mtext
index = 0
numberWrong = 0
checkfieldsyntax ("mat"&i&j)
studans =
element = 0
g((i-1)*
"m1")*(
g((k-1)*col+j)
"m2")
firsttry
4][2]=
4][1]=
question
" = numberRight
4][1]
"Correct"
f"Next
"Quit exercise"
4][1]=
4][1]
"Incorrect:
the highlighted value indicates
incorrect
f"Try Again"
"Show Answer"
Zlues
Delements"
tidyup
B"QuitEx"
"matrix"
genquestion
"ex4"
enabled
B"enter" =
"explan" = ""
"ansback"
"mg1"
H-100,0
"mg2"
H-760,0
--
col-1
(i-1)*
"m2" = "-"
--
--
--
"+"&(
"m2")
--
B("c"&i&j)
"exback"
bds =
B"c11"
xpos =
ypos =
bot =
B("c"&
&"1")
drawbracket
!"a9" =
!"a10" =
B("c1"&
!"a11" =
!"a12" =
g((i-1)*
"m1")*(
g((k-1)*
"m2")
B("c"&i&j) =
firsttry
4][2]=
4][1]=
question
" = numberRight
matex4
matex3
sbmflashredffw
quitmc
exback
multcondition
quitex
leavepage
sbmflashredffw f,g,h,
origcolourf
origcolourg
origcolourh
exback
Bquitex
multcondition
Bquitmc
"~"`'X2
choice
count
Multiplication of Matrices
Not all matrices can be multiplied together.
Matrices A (of order nxm) and B (of order pxq) can be multiplied to form the product AB, if and only if the condition m = p holds. The resulting matrix will be of order nxq.m = p. The resulting matrix will be of order nxq........
multiply
buttonup
multiply
Matrix Algebra
board and exercise
3 of 5
Page 9 of 12w
answer(2)
answer(1)
MULTIPLICATIVE CONDITION
Exercise 3
.&, 8
forward
multcondition
order
pageno
inner
status
buttonup
.&+ +E
mouseenter
.&+ +E
mouseleave
status=""
"pageno"
"inner"
"multcondition"
order
180,50,100
180,50,100
" = "More"
choice
This multiplicative condition is best illustrated by way of a diagram.
Click here to start
multcondition
noshow
choice
z*c J
A B = AB
order
order nxm mxp nxp
inner
outer
textinner
The product of the matrices A and B only exists if the number of columns of A equals the number of rows of B.
textouter
The product of the matrices A and B has order nxp, i.e it has the same number of rows as the matrix A and the same number of columns as the matrix B...
quitmc
forward
textouter
showmewhy
Click red text, green board and exercise
multcondition
lookslike
page3
outer
status
buttonup
"multcondition"
"outer"
"textouter"
"page3"
"lookslike"
"showmewhy"
status="Click
board
exercise"
Remove
forward
pageno
1 of 333333
page3
page3
For an example of the multiplicative condition consider the
matrices
A = , B =
why ?
2 7 8 1
-1 1 -1 3
2 1 8 3
3 0 1 4
5 -1 7 2
2 7 8 1
-1 1 -1 3
X9p8U9
2 7 8 1
-1 1 -1 3
*;B:';
4 1
2 2
9 -1
3 77777777777777
2 7 8 1
-1 1 -1 3
Does the product A and B exist?
Show me why ????????
lookslike
buttonup
"lookslike"
showmewhy
multiply
buttonup
"multiply"
"showmewhy"
lookslike
buttonup
The product of A and B, say C,
does exist and looks like C = .
Click to remove
2 7 8 1
-1 1 -1 3
* *
* *
* *
2 7 8 1
-1 1 -1 3
showmewhy
buttonup
A B = AB
order 3x4 4x2 3x2
outer
Since the values in this inner bracket are the same the matrices can be multiplied.
This outer bracket shows the order of the resulting matrix, i.e 3x2.
Click to remove
exback
exback
Consider the matrix product:
where B is replaced in turn by each of the matrices below.
Click onto the check box (or the matrix) for each matrix where the product exists. Then click the Enter key.
Enter
QuitEx
Remove
smile1
buttonup
smile4
buttonup
smile3
buttonup
smile2
buttonup
frown1
frown4
frown3
frown2
Exercise 3333
Matrix Multiplication Conditionn
question
Question 1447
score
Score: 0 out of 0344
MULTIPLY
buttonup
choice
In the product AB, B is said to be pre-multiplied by A, and A is post-multiplied by B.
Click to removeee
blayer
Click red text, green board and exercise
mes New Roman
Times New Roman
OtIKsp
OtIKsp
dJGvvzALJfF
Symbol
Symbol
Times New Roman
exinuse
Arial
matratio
fntsize
fntwidth
.TBK"
Arial
Times New Roman
System
Arial
Arial
all Fonts
Symbol
Times New Roman
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Wingdings
Times New Roman
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J z
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timesnewroman
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Page id 13 of Book "C:\TB30\CAL\INDEF1.TBK"
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Times New Roman
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0,numrights
tbkex1
urier
tbkex
urier
Times New Roman
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Times New Roman
System
Symbol
Times New Roman
Wingdings
Times New Roman
suisys.tbk
entersystem
matmultflag
animateflag
arraypos
11 12 21 22 31 32
enterbook
"suisys.tbk"
4animateflag,matmultflag
garraypos[6]
fill
f"11 12 21 22 31 32"
g] order
0 -- used
handler
matrix addition animation
0 --
,multiplication
Introduction to Matrices
general
Page id 155 of Book "D:\TB30\TLTPCAL\OLDEX\MAT1.TBK"
CDBSE&File
&Open... Ctrl+O
&Save Ctrl+S
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&Print Pages... Ctrl+P
printpages
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printreport
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paste
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selectall
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INTRODUCTION TO MATRICES
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Introduction
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3l7R9
A MATRIX, eg.
is a set of elements (usually numbers) arranged in a rectangular array.
MATRICES is the plural of MATRIX.
ntroduce some definitions and properties of matrices, we will look at a few examples.
single symbol (usually a capital letter).
To introduce some definitions and properties of matrices, we will look at a few examples.
ices.
Only after you have learnt the "language" of matrices can you progress to
actually manipulating them, which you do later in the Matrix Algebra section.ing vocabulary in order to speak a language.
Only after you have learnt the "language" of matrices can you progress to
actually manipulating them, which you do in the "Matrix Algebra" section
in the next module......................ry in order to speak a language.
Only after you have learnt the "language" of matrices can you progress to
actually manipulating them, which you do in the "Matrix Algebra" section
in the next module.............
Introduction
1 of 1
Page 1 of 12
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This module will take you through an introduction to matrices. Vectors are also introduced, but for a more detailed explanation see the separate Vectors Module.y First and Second Order O.D.E.s are considered here.
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The two main sections of this module cover
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Definitions
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1. Notation and Definitions
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2. Matrix algebra
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Firstly, a number of definitions are given, these may seem tedious but they are important. In particular, you will need to know about the order of a matrix and suffix notation.
2 of 4
page4
Summary of Content
Definitions
Matrix addition
Matrix multiplication
n the plane
ent O.D.E.s
Good Luck!
4 of 4'(
algebra
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Matrix algebra covers the addition, subtraction and multiplication of matrices. However, matrices are never divided.....
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Matrices enable a collection of many numbers to be considered as a single object, denoted by a single symbol (usually a capital letter).
With this convenient notation matrices prove useful in a variety of applications.
of many numbers to be considered as a single object, denoted by a single symbol (usually a capital letter).
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There are several words that we need to introduce to explain all the properties of matrices.
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A vector is a special type of matrix that has either only one column or one row, e.g.
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Matrices can be used to represent any set of numerical data which could be written in the form of a table.
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Go to page 2 of this book for specific examples.
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applications
Matrices are useful in analysing many practical problems. Click onto the numbered boxes to see some illustrations. introduce some definitions and properties of matrices, we will look at a few examples.
The numbers in the table can be written in matrix form by stripping out the other details and enclosing the data with brackets.
This is a "five-by-four" matrix (written 5x4) since it has five rows and four columns.
3 2 4 3
2 2 6 1
0 4 0 5
2 7 8 1
10 5 4 0
5 rows
4 columns
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3 2 4 3
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10 5 4 0 0 4 0 4 0 4 0 6 1
Liberal 0 4 0 5
Acid-House 2 7 8 1
Tupperware 10 5 4 0
text2
This is a "five-by-four" matrix (written 5x4) since it has five rows and four columns.
d enclosing the data with brackets.
This is a "five-by-four" matrix (written 5x4) since it has five rows and four columns.
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1 1 1
0 0 2
1 0 0
7 8 1
10 5 4 0
Consider this map of three places A, B and C and their connecting roads.
It has associated with it a CONNECTIVITY TABLE showing the number of direct paths between the places.
So, remembering to go FROM the columns TO the rows, a CONNECTIVITY MATRIX can describe such a network of routes:
1 1 1
To : B
0 0 2
C
1 0 0
connectivity table shows the number of paths between places on the map. on the map.............. places on the map.
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simtext1
Suppose we have to solve three simultaneous equations in three unknowns, say x, y and z.
It is not the VARIABLES which are important in these equations, but rather the NUMBERS multiplying them.
You can see that the above equations are summarised by a 3x4 matrix.
This is often referred to as an AUGMENTED MATRIX, since the numerical entries from the left-hand sides of the equations have been augmented with those from the right-hand sides.
simtext3
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7x + 4y - z = 2
3x - 2y + 4z = -6
- x + y - z = 3
z = 3
simmat5
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7 4 -1 2
3 -2 4 -6
-1 1 -1 3
2 7 8 1
-1 1 -1 3
simmat1
7 4 -1
3 - 2 4
-1 1 - 1 3 3 3
2 7 8 1
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simmat2
3 1 - 1 3 1 - 1 3 1 - 1 3
2 7 8 1
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Separate the numbers
simmat3
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-1 1 - 1 3 3 3
2 7 8 1
-1 1 -1 3
simmat4
3 1 - 1 3 1 - 1 3 1 - 1 3
2 7 8 1
-1 1 -1 3
simtext2
You can see that the above equations are summarised by a 3x4 matrix.
This is often referred to as an AUGMENTED MATRIX, since the numerical entries from the left-hand sides of the equations have been augmented with those from the right-hand sides.
d by a 3x4 matrix.
This is often referred to as an AUGMENTED MATRIX, since the numerical entries from the left-hand sides of the equations have been augmented with those from the right-hand sides.
simtext3
The Advanced Matrices module tells you how to solve systems of equations using matrix methods.
ctobline
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hTiX|
Addition of Matrices
Click on the buttons below to find out about matrix addition.
Matrix Algebra
Click on numbered buttons
2 of 5
Page 8 of 12
answer(2)
answer(1)
MATRIX ADDITION
Exercise 2cise 3e 2
table
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When can matrices be added together?
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addition
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Some important rules traction ?dition
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then
A + B = + =
and B =
then
A + B = + =
ilar result holds for matrices of other orders.
s for matrices of other orders.
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2 7 8 1
-1 1 -1 3
2 7 8 1
-1 1 -1 3
5 7
4 1
-2 333
2 7 8 1
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3 6
5 0
6 233
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2 7 8 1
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Consider the matrices
A = and B =
A + B = + =
ilar result holds for matrices of other orders.
s for matrices of other orders.
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2 7 8 1
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addition
For example, the sum of two general 2x2 matrices, A and B, is
A + B = +
= .
.
es of other orders.
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addition
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10 5 4 0
b b
b b
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`cxb]c
a + b a + b
a + b a + b
4 0
10 5 4 0
Two matrices are added together by summing the corresponding elements of the matrices.
+ = .
es of other orders.
order
Thus, if
same ORDER may be added. Thus, if
*lBk'l
4 3
2 -1
4 0
4 0
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8 -2
0 6
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4 0
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, B ====
then A + B =
12 1
2 5
4 0
4 0
10 5 4 0
same ORDER may be added. Thus, if
4 3
2 -1
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, B ====
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4 0
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Why not?
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A order 2x2
B order 2x3 e}
subtraction
Matrix SUBTRACTION is performed in a similar way to matrix addition.
A - B = - = = . . = = . =========================
For example, if A = , B =
then
A - B = - =
= . - = = . =========================
0 3
7 5
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0 1
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remsubtraction
subtraction
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rules
Since matrix addition is just a case of adding corresponding elements in the matrices, some important rules follow:
click on the equations for more information = =
A - B = - = ===========
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associative
commutative
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associative
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A + (B+C) = (A+B) + C AAAAAA
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addidentity
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A + 0 = 0 + A = A
Click on the blue equations opposite for information about each rule.
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A = , B =
C = A + B
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Exercise 222222
Matrix Additioncation condition
question
Question 18
order2
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click to remove
score
Score: 0 out of 077
control
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C = A+B =
ansback
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A+B = + = =
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addidentity
If you recall the null (zero) matrix, 0 say, you will see that
A + 0 = 0 + A = A
for any matrix A. Thus 0 is the identity for matrix addition.
This should not be confused with I = , the identity for matrix
multiplication.
addidentity
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Remove
associative
Matrices are associative under addition, i.e. the brackets may be ignored when adding matrices together.
However, matrix subtraction is not associative.
associative
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commutative
Matrices are commutative under addition, i.e. the same result is obtained irrespective of the order in which the sum is written.
However, matrix subtraction is not commutative (as with numbers the order matters).
commutative
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orderequal
A and B both
order 2x2
matex4
LTS*T
WXX.Y
AB
(AB)C = A(BC)
AI = IA = A
Click on the exercise book to
start the exercises..
ee that
AI = IA = A
for any matrix A. Thus I is the IDENTITY for matrix MULTIPLICATION.
for any matrix A. Thus I is the IDENTITY for matrix MULTIPLICATION.
. Thus I is the IDENTITY for matrix MULTIPLICATION.
Thus I is the IDENTITY for matrix MULTIPLICATION.
ee matrices A, B and C together, we find that
(AB)C = A(BC)
so that matrix multiplication is, like matrix addition, ASSOCIATIVE.
If you recall the identity matrix I in MATRICES(1), you will see that
AI = IA = A
for any matrix A. Thus I is the IDENTITY for matrix MULTIPLICATION.
ee that
AI = IA = A
for any matrix A. Thus I is the IDENTITY for matrix MULTIPLICATION.
commutative
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"commutative"
associative
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"associative"
identity
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"identity"
Matrix Algebra
Click on blue text and exercise book
5 of 5
Page 11 of 12
Click on the blue equations opposite to learn more about the rules of matrix multiplication.
answer(2)
answer(1)
."%#X
MATRIX MULTIPLICATION
Exercise 4 4e 2
exback
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A = , B =
C = A * B
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Exercise 444
Matrix Multiplicationnnn condition
question
Question 1411
order2
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klogorder
buttonup
logorder
tabexplan
buttonup
The Tab key is usually found to the left of the character Q on the keyboard. The key may be indicated by the symbols
click to remove
score
Score: 0 out of 044
associative
If 3 matrices A,B and C are multiplied together then the brackets may safely be ignored, i.e. matrix multiplication is, like matrix addition, associative.
ubtraction is not associative. n adding matrices together.
However, matrix subtraction is not associative. ive.
associative
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commutative
Matrix multiplication is, in general, not commutative, i.e. the same result is not obtained irrespective of the order in which the product is written.
In fact, sometimes only one of the products AB and BA will be defined.
subtraction is not commutative.
commutative
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identity
Recall the identity matrix, I.
Multiplication of any matrix A by I leaves A unchanged.
I is the identity matrix under multiplication.
Remind me what the identity looks like !
This should not be confused with I, the identity matrix under multiplication.
buttonup
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identity
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I = (of order 3x3))
4FLE1F
2 7 8 1
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1 0 0
0 1 0
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2 7 8 1
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matrix
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Use the Tab key (or mouse) to move from element to element. Enter the appropriate value for each element. Once all the elements have been entered click the Enter key.......
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C = A*B =
ansback
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A*B = * =
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Click onto elements of the matrix product to find out how they were calculated.
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matrix algebar
matrix algebra
identity looks like !
This should not be confused with I, the identity matrix under multiplication.
(.txt)