Chapter 3.6, Problem 23PPS

To determine

To find: Whether the matrix product is defined and also state the dimension for the product of $\left[\begin{array}{cc}-3& -7\\ -2& -1\end{array}\right]\cdot \left[\begin{array}{cc}4& 4\\ 9& -3\end{array}\right]$ .

Answer to Problem 23PPS

The matrix product is defined and the dimension of $\left[\begin{array}{cc}-75& 9\\ -17& -5\end{array}\right]$ is. $2\times 2$ .

Explanation of Solution

Given:

The expression is $\left[\begin{array}{cc}-3& -7\\ -2& -1\end{array}\right]\cdot \left[\begin{array}{cc}4& 4\\ 9& -3\end{array}\right]$ .

Calculation:

The first matrix is of the order $2\times 2$ and the second matrix is of the order $2\times 2$ .

The product of two matrices is defined as the number of columns of the first matrix is equal to the number of rows of the second matrix.

The order of matrices $\left[\begin{array}{cc}-3& -7\\ -2& -1\end{array}\right]\cdot \left[\begin{array}{cc}4& 4\\ 9& -3\end{array}\right]$ as follows:

  $\begin{array}{l}\text{Matrix}1\text{st}\text{Matrix}2\text{nd}\\ 2\times 22\times 2\\ \Downarrow \\ \text{ 2}\times \text{2}\end{array}$

Therefore, the product is defined and dimension is $2\times 2$ .

Multiply the matrices to obtain the product:

  $\begin{array}{c}\left[\begin{array}{cc}-3& -7\\ -2& -1\end{array}\right]\cdot \left[\begin{array}{cc}4& 4\\ 9& -3\end{array}\right]=\left[\begin{array}{cc}-12-63& -12+21\\ -8-9& -8+3\end{array}\right]\\ =\left[\begin{array}{cc}-75& 9\\ -17& -5\end{array}\right]\end{array}$

Thus, the matrix product is defined and the dimension of $\left[\begin{array}{cc}-75& 9\\ -17& -5\end{array}\right]$ is. $2\times 2$ .