Chapter 11, Problem 10CT

To determine

The value of the expression is $BA$ where $A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right],B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right],C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]$.

Answer to Problem 10CT

Solution:

The value of the expression $BA$ is $\left[\begin{array}{cc}16& 17\\ 3& -10\end{array}\right]$.

Explanation of Solution

Given information:

The matrices, $A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right],B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right],C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right]$.

Explanation:

Consider an expression $BA$,

Here is the product of a $2\times 3$ matrix and a $3\times 2$ matrix.

The number of columns in the first matrix is the same as the number of rows in the second matrix, the operation can be performed.

$\Rightarrow BA=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& 2\end{array}\right]$

$\Rightarrow BA=\left[\begin{array}{cc}1\cdot 1+\left(-2\right)\cdot 0+5\cdot 3& 1\left(-1\right)+\left(-2\right)\cdot \left(-4\right)+5\cdot 2\\ 0\cdot 1+3\cdot 0+1\cdot 3& 0\cdot \left(-1\right)+3\cdot \left(-4\right)+1\cdot 2\end{array}\right]$

$\Rightarrow BA=\left[\begin{array}{cc}1+0+15& -1+8+10\\ 0+0+3& 0-12+2\end{array}\right]$

$\Rightarrow BA=\left[\begin{array}{cc}16& 17\\ 3& -10\end{array}\right]$

Therefore, the value of the expression $BA$ is $\left[\begin{array}{cc}16& 17\\ 3& -10\end{array}\right]$.