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%TCIDATA{<META NAME="Title" CONTENT="Standard Operations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:08:39 1997}
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\begin{document}


\section{Standard Operations\label{Standard matrix operations}}

You can perform standard operations on matrices, such as addition,
subtraction, and multiplication, by evaluating expressions entered in
natural notation.

\subsection{\ Matrix Addition and Scalar Multiplication}

You add two matrices of the same dimension by adding corresponding entries.
The numbers or other expressions used as matrix entries are called \textsl{%
scalars}. You multiply a scalar with a matrix by multiplying every entry of
the matrix by the scalar. You can do matrix addition and multiplication and
other operations with scalars and matrices by choosing \textsf{Evaluate}.
Place the insertion point anywhere inside the expression.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] +\left[ 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right] =\left[ 
\begin{array}{cc}
6 & 8 \\ 
12 & 10
\end{array}
\right] \smallskip $
\end{quotation}

Note that the sum appears with the same brackets as the original matrices.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
a_{11} & a_{12} \\ 
a_{21} & a_{22}
\end{array}
\right) +\,\left( 
\begin{array}{cc}
b_{11} & b_{12} \\ 
b_{21} & b_{22}
\end{array}
\right) =\,\left( 
\begin{array}{cc}
a_{11}+b_{11} & a_{12}+b_{12} \\ 
a_{21}+b_{21} & a_{22}+b_{22}
\end{array}
\right) $

\ 

$a\left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] =\,\left[ 
\begin{array}{cc}
a & 2a \\ 
4a & 3a
\end{array}
\right] $

\ 

$a\left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] -b\left[ 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right] =\,\left[ 
\begin{array}{cc}
a-5b & 2a-6b \\ 
4a-8b & 3a-7b
\end{array}
\right] \smallskip $
\end{quotation}

\strut

\subsection{\ \label{Matrix product}Inner Products and Matrix Multiplication}

The product of a $1\times n$ matrix with an $n\times 1$ matrix (the product
of two vectors) produces a scalar called the 
\index{Inner product@Inner product}\label{Inner product (2)}\textsl{inner
product} (sometimes called the \textsl{dot product}) 
\index{Dot product@Dot product}\label{Dot product (2)}of the two vectors. 
\index{Vector@Vector!inner product@inner product}\label%
{Vector, inner product}The \textsl{matrix product} of an $m\times k$ matrix
with a $k\times n$ matrix is an $m\times n$ matrix obtained 
\index{Matrix product@Matrix product}by taking inner products of rows and
columns, the $ij$th entry of the product $AB$ being the inner product of the 
$i$th row of $A$ with the $j$th column of $B$.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
a & b
\end{array}
\right) \left( 
\begin{array}{c}
c \\ 
d
\end{array}
\right) =\,ac+bd$

\ 

$\left( 
\begin{array}{cc}
a & b \\ 
u & v
\end{array}
\right) \left( 
\begin{array}{c}
c \\ 
d
\end{array}
\right) =\,\left( 
\begin{array}{c}
ac+bd \\ 
uc+vd
\end{array}
\right) $

\ 

$\left( 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right) \left( 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right) =\,\left( 
\begin{array}{cc}
21 & 20 \\ 
44 & 45
\end{array}
\right) $

\ 

$\left[ 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right] ^{3}=\,\left[ 
\begin{array}{cc}
941 & 942 \\ 
1256 & 1255
\end{array}
\right] \smallskip $
\end{quotation}

To put an exponent on a matrix, place the insertion point immediately on the
right of the matrix, click%
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choose \textsf{Insert + Superscript}, and type the exponent in the input box.

\subsection{\ \label{Identity matrix}Identity and Inverse Matrices}

The $n\times n$ \textsl{identity} matrix $I$ has ones down the main diagonal
(upper-left corner to lower-right corner) and zeroes elsewhere. 
\index{Matrices@Matrices!identity@identity}The $3\times 3$ identity matrix,
for example, is 
\[
I=\left[ 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right] 
\]

The \textsl{inverse} 
\index{Matrices@Matrices!inverse@inverse}\label{Matrices, inverse} 
\index{Inverse@Inverse!matrix@matrix}\label{Inverse matrix}of an $n\times n$
matrix $A$ is an $n\times n$ matrix $B$ satisfying $AB=I$. To find the
inverse of an invertible matrix $A$, enter $A$ with ``$-1$'' as a
superscript and apply \textsf{Evaluate}. (As an alternative, leave the
insertion point anywhere inside the matrix $A$, and from the \textsf{Matrices%
} submenu, choose \textsf{Inverse}.)

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right) ^{-1}\,=\,\left( 
\begin{array}{rr}
-%
\dfrac{7}{13}\vspace{6pt} & \dfrac{6}{13} \\ 
\dfrac{8}{13} & -\dfrac{5}{13}
\end{array}
\right) \smallskip $
\end{quotation}

\strut

To check that this matrix satisfies the defining property, evaluate the
product.

\begin{quotation}
$\left( 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right) \,\left( 
\begin{array}{rr}
-\dfrac{7}{13}\vspace{6pt} & \dfrac{6}{13} \\ 
\dfrac{8}{13} & -\dfrac{5}{13}
\end{array}
\right) =\,\left( 
\begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}
\right) \smallskip $
\end{quotation}

\strut

The operation \textsf{Evaluate Numerically} gives you a numerical
approximation of the inverse. The accuracy of this numerical approximation
depends on properties of the matrix, as well as on the choices you have made
in the \textsf{Settings} menu.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right) ^{-1}=\,\left( 
\begin{array}{rr}
-.53846 & .46154 \\ 
.61538 & -.38462
\end{array}
\right) \smallskip $
\end{quotation}

\strut

Checking the product of a matrix with its inverse gives you an idea of the
degree of accuracy of the approximation.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right) \left( 
\begin{array}{rr}
-.53846 & .46154 \\ 
.61538 & -.38462
\end{array}
\right) =\,\left( 
\begin{array}{rr}
.99998 & -.00002 \\ 
-.00002 & .99998
\end{array}
\right) \smallskip $
\end{quotation}

\strut

Since $\left( A^n\right) ^{-1}=\left( A^{-1}\right) ^n$, you can compute
negative powers of invertible matrices.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right) ^{3}=\left( 
\begin{array}{cc}
941 & 942 \\ 
1256 & 1255
\end{array}
\right) $

\qquad

$\left( 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right) ^{-3}=\left( 
\begin{array}{cc}
-\frac{1255}{2197}\vspace{6pt} & \frac{942}{2197} \\ 
\frac{1256}{2197} & -\frac{941}{2197}
\end{array}
\right) $

\ 

$\left( 
\begin{array}{cc}
941 & 942 \\ 
1256 & 1255
\end{array}
\right) \left( 
\begin{array}{cc}
-\frac{1255}{2197}\vspace{6pt} & \frac{942}{2197} \\ 
\frac{1256}{2197} & -\frac{941}{2197}
\end{array}
\right) =\left( 
\begin{array}{cc}
1 & 0 \\ 
0 & 1
\end{array}
\right) \smallskip $
\end{quotation}

\strut

The preceding product demonstrates that $A^{-3}=\left( A^3\right) ^{-1}$.

The $m\times n$ matrix with every entry equal to zero is the \textsl{%
identity for addition}; that is, for any $m\times n$ matrix $A$, 
\[
A+0=0+A=A 
\]
and the \textsl{additive inverse} of a matrix $A$ is the matrix $\left(
-1\right) A$.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
a_{1,1} & a_{1,2} \\ 
a_{2,1} & a_{2,2} \\ 
a_{3,1} & a_{3,2}
\end{array}
\right] +\left[ 
\begin{array}{cc}
-a_{1,1} & -a_{1,2} \\ 
-a_{2,1} & -a_{2,2} \\ 
-a_{3,1} & -a_{3,2}
\end{array}
\right] =\,\left[ 
\begin{array}{cc}
0 & 0 \\ 
0 & 0 \\ 
0 & 0
\end{array}
\right] $\ $\smallskip $
\end{quotation}

\strut

\subsection{\ \label{Matrix Values}Polynomials with Matrix Values}

\strut

\begin{example}
A polynomial expression, such as $x^{2}-5x-2$, can be evaluated at a matrix.

\begin{itemize}
\item  Leave the insertion point in the expression $x=\left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] $, and from the \textsf{Define }submenu, choose \textsf{New
Definition.}

\item  Apply \textsf{Evaluate} to the polynomial .
\end{itemize}
\end{example}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$x^{2}-5x-2=\,\left[ 
\begin{array}{rr}
2 & -2 \\ 
-4 & 0
\end{array}
\right] $

$x^{2}-5x-2x^{0}=\allowbreak \left[ 
\begin{array}{cc}
2 & -2 \\ 
-4 & 0
\end{array}
\right] \smallskip $
\end{quotation}

\strut

You can also define the function $f(x)=x^{2}-5x-2$ and apply \textsf{Evaluate%
} (twice) to get the following.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate, Evaluate}
\end{quote}

\begin{quotation}
$f\left( \left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] \right) =\allowbreak \left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] ^{2}-5\left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] -2=\allowbreak \left[ 
\begin{array}{cc}
2 & -2 \\ 
-4 & 0
\end{array}
\right] \smallskip $
\end{quotation}

\strut

The expression\textsf{\ }$-5\left[ 
\begin{array}{cc}
1 & 2 \\ 
4 & 3
\end{array}
\right] -2$ is not, strictly speaking, a proper expression. However, when
evaluated by in \textsl{Scientific Notebook}, the $2$ is interpreted in this
context as $\left[ 
\begin{array}{cc}
2 & 0 \\ 
0 & 2
\end{array}
\right] $, or twice the $2\times 2$ identity matrix.

\strut

\subsection{\ Operations on Matrix Entries\label{Matrix Entries}}

To operate on one entry of a matrix, select the entry, and choose the
operation while holding down the \textsc{ctrl }key. That will perform the
operation in place, leaving the rest of the matrix unchanged. Because you
are in a word-processing environment, you can edit individual entries (just
click in the input box and then edit) and apply other word-processing
features to entries, such as copy and paste or click and drag.

Many of the operations on the \textsf{Maple} menu operate directly on the
entries when applied to a matrix, as can be seen from the following examples.

\begin{quote}
$\blacktriangleright $ \textsf{Factor}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
5 & 6 \\ 
8 & 7
\end{array}
\right] =\left[ 
\begin{array}{cc}
5 & 2\times 3 \\ 
2^{3} & 7
\end{array}
\right] \smallskip $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
\vspace{8pt}\frac{d}{dx}\sin x & \int 6x^{2}dx \\ 
\frac{d^{2}}{dx^{2}}\ln x & x+3x
\end{array}
\right] =\,\left[ 
\begin{array}{cc}
\cos x & 2x^{3} \\ 
-\frac{1}{x^{2}} & 4x
\end{array}
\right] \smallskip $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
\sin ^{2}\pi & e \\ 
\ln 5 & x+3x
\end{array}
\right] =\,\left[ 
\begin{array}{cc}
0 & 2.7183 \\ 
1.6094 & 4.0x
\end{array}
\right] \smallskip $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Combine + Trig Functions}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
\sin ^{2}x+\cos ^{2}x & 6x^{2} \\ 
4\sin 4x\cos 4x & \sin x\cos y+\sin y\cos x
\end{array}
\right] =\,\left[ 
\begin{array}{cc}
1 & 6x^{2} \\ 
2\sin 8x & \sin \left( x+y\right)
\end{array}
\right] \smallskip $
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\frac{d}{dx}\left[ 
\begin{array}{cc}
x+1 & 2x^{3}-3 \\ 
\sin 4x & 3\sec x
\end{array}
\right] =\left[ 
\begin{array}{cc}
1 & 6x^{2} \\ 
4\cos 4x & 3\sec x\tan x
\end{array}
\right] \smallskip $
\end{quotation}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Working With Matrices}{}{}{DM8-1.tex#Matrix Algebra}

\item  \hyperref{Row Operations and Echelon Forms}{}{}{DM8-3.tex#Matrices,
row operations}

\item  \hyperref{Systems of Linear Equations}{}{}{DM8-4.tex#Systems of
equations}

\item  \hyperref{Matrix Equations}{}{}{DM8-4.tex#Matrix equations}

\item  \hyperref{Matrix Operators}{}{}{DM8-5.tex#Trace}

\item  \hyperref{Polynomials and Vectors Associated with a Matrix}{}{}{%
DM8-6.tex#Characteristic, polynomial}

\item  \hyperref{Vector Spaces Associated with a Matrix}{}{}{DM8-7.tex#Row
space}

\item  \hyperref{Special Forms of Matrices}{}{}{DM8-8.tex#Smith normal form}

\item  \hyperref{Exercises and Solutions}{}{}{DM8-9.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
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DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Matrices}{}{}{%
DMIndex.tex#Matrices}, \hyperref{Vector}{}{}{DMIndex.tex#Vector} }

\strut

\end{document}
