Inner Products and Matrix Multiplication

The product of a 1×n matrix with an n×1 matrix (the product of two vectors) produces a scalar called the inner product (sometimes called the dot product) of the two vectors. The matrix product of an m×k matrix with a k×n matrix is an m×n matrix obtained by taking inner products of rows and columns, the ijth entry of the product AB being the inner product of the ith row of A with the jth column of B.

$\blacktriangleright$ Evaluate

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

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

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

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

To put an exponent on a matrix, place the insertion point immediately on the right of the matrix, clickitbpF0.3009in0.3009in0.0701insupscrip.wmf or choose Insert + Superscript, and type the exponent in the input box.