Chapter 11.4, Problem 27AYU

To determine

The product of the matrix is defined and if so, find the product.

Answer to Problem 27AYU

Solution:

The product is DEFINED and is $=\left[\begin{array}{ll}5\hfill & 14\hfill \\ 9\hfill & 16\hfill \end{array}\right]$

Explanation of Solution

Given:

  $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& -1& 4\end{array}\right]\text{ \&}B=\left[\begin{array}{cc}1& 2\\ -1& 0\\ 2& 4\end{array}\right]$

Calculation:

It can be observed that $A$ is a $2\text{x}3\text{ \&B}$ is $3\text{x}$ 2. The number of columns in $A$ is equal to the number of rows in $\text{B},$ so the product AB is defined.

  $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& -1& 4\end{array}\right]$

  $B=\left[\begin{array}{cc}1& 2\\ -1& 0\\ 2& 4\end{array}\right].$

  $=\left[\begin{array}{cc}(1\times 1)+(2\times -1)+(3\times 2)& (1\times 2)+(2\times 0)+(3\times 4)\\ (0\times 1)+(-1\times -1)+(4\times 2)& (0\times 2)+(-1\times 0)+(4\times 4)\end{array}\right]$

  $=\left[\begin{array}{ll}1-2+6\hfill & 2+0+12\hfill \\ 0+1+8\hfill & 0+0+16\hfill \end{array}\right]$

  $=\left[\begin{array}{ll}5\hfill & 14\hfill \\ 9\hfill & 16\hfill \end{array}\right]$