Chapter 16.3, Problem 2WE

To determine

To find product of the matrices.

Answer to Problem 2WE

The product of given matrices is:

  $\left[-16\right]$

Explanation of Solution

Given:

The matrices:

  $\left[\begin{array}{ccc}0& -3& 4\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -4\end{array}\right]$

Concept Used:

If $A=\left[a_{ij}\right]$ is an $m\times n$ matrix and $B=\left[b_{ij}\right]$ is an $n\times p$ matrix, then the product AB is an $m\times p$ matrix given by $AB=\left[c_{ij}\right]$ , where $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}\mathrm{...}+a_{in}{b}_{nj}$ .

Calculation:

First check if this product is possible.

Note that here number of columns of first matrix is 3 and number of rows of 2^nd matricesis 3 which are same. Thus this product is possible.

Now find this product $AB=\left[c_{ij}\right]$ using formula $c_{ij}=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}\mathrm{...}+a_{in}{b}_{nj}$ .

  $\begin{array}{c}\left[\begin{array}{ccc}0& -3& 4\end{array}\right]\left[\begin{array}{c}1\\ 0\\ -4\end{array}\right]=\left[0\times 1+\left(-3\right)\left(0\right)+4\left(-4\right)\right]\\ =\left[0+0-16\right]\\ =\left[-16\right]\end{array}$

Thus, product of given matrices is:

  $\left[-16\right]$