Chapter 3.6, Problem 25PPS

To determine

To find: Whether the matrix product is defined and also state the dimension for the product of $\left[\begin{array}{ccc}-1& 0& 6\\ -4& -10& 4\end{array}\right]\cdot \left[\begin{array}{cc}5& -7\\ -2& -9\end{array}\right]$ .

Answer to Problem 25PPS

The matrix product of $\left[\begin{array}{ccc}-1& 0& 6\\ -4& -10& 4\end{array}\right]\cdot \left[\begin{array}{cc}5& -7\\ -2& -9\end{array}\right]$ is undefined.

Explanation of Solution

Given:

The expression is $\left[\begin{array}{ccc}-1& 0& 6\\ -4& -10& 4\end{array}\right]\cdot \left[\begin{array}{cc}5& -7\\ -2& -9\end{array}\right]$ .

Calculation:

The first matrix $\left[\begin{array}{ccc}-1& 0& 6\\ -4& -10& 4\end{array}\right]$ is of the order $2\times 3$ and the second matrix $\left[\begin{array}{cc}5& -7\\ -2& -9\end{array}\right]$ is of the order $2\times 2$ .

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

The order of matrices $\left[\begin{array}{ccc}-1& 0& 6\\ -4& -10& 4\end{array}\right]\cdot \left[\begin{array}{cc}5& -7\\ -2& -9\end{array}\right]$ as follows:

  $\begin{array}{l}\text{Matrix}1\text{st}\text{Matrix}2\text{nd}\\ 2\times 32\times 2\\ \Downarrow \\ \text{ Impossible}\end{array}$

Therefore, the product is not defined as the number of columns of the first matrix is not equal to the number of rows of the second matrix.

Thus, the matrix product of $\left[\begin{array}{ccc}-1& 0& 6\\ -4& -10& 4\end{array}\right]\cdot \left[\begin{array}{cc}5& -7\\ -2& -9\end{array}\right]$ is undefined.