Chapter 7, Problem 7RE

Solution

a.

To calculate: The matrix multiplication $AB$ by using the definition of the matrix multiplication.

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

The multiplication is $\left[\begin{array}{ccc}1& 2& -3\\ 2& -3& 4\\ -2& 1& -1\end{array}\right]$ .

Given Information:

The matrix are,

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

Formula Used:

The matrix multiplication is possible if and only if order of the first matrix column equal to the order of the second matrix row.

Calculation:

Consider the given matrix,

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

It can be observed that the number of column of matrix $A$ is equal to the number of row of the matrix $B$ . As $A$ have 3 column and $B$ have 3 rows.

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

Hence, the multiplication is $\left[\begin{array}{ccc}1& 2& -3\\ 2& -3& 4\\ -2& 1& -1\end{array}\right]$ .

b.

To calculate: The matrix multiplication $BA$ by using the definition of the matrix multiplication.

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

The multiplication is $\left[\begin{array}{ccc}-3& 2& 4\\ 2& 1& -3\\ 1& -2& -1\end{array}\right]$ .

Given Information:

The matrix are,

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

Formula Used:

The matrix multiplication is possible if and only if order of the first matrix column equal to the order of the second matrix row.

Calculation:

Consider the given matrix,

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

It can be observed that the number of column of matrix $B$ is equal to the number of row of the matrix $A$ .

Now, find the value of $BA$ .

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

Hence, the obtained matrix after multiplication is $\left[\begin{array}{ccc}-3& 2& 4\\ 2& 1& -3\\ 1& -2& -1\end{array}\right]$ .