Chapter 16.3, Problem 19WE

To determine

To find: the $0_{2\times 2}A$ and $A{.0}_{2\times 2}$ using the given matrices.

Answer to Problem 19WE

  $0_{2\times 2}\times A=A\times 0_{2\times 2}=0_{2\times 2}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right].$

Explanation of Solution

Given:

Matrices $A$ :

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

Matrices $B$ :

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

Matrices $C$ :

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

Concept used:

Zero matrix multiply by any number or matrix is zero.

Two multiply the two matrices column of the first matrices must equals to the raw of the second matrices.

  $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\times \left[\begin{array}{cc}{b}_{11}& b_{12}\\ b_{21}& b_{22}\end{array}\right]=\left[\begin{array}{cc}{a}_{11}{b}_{11}+a_{12}{b}_{21}& a_{11}{b}_{12}+a_{12}{b}_{22}\\ a_{21}{b}_{11}+a_{22}{b}_{21}& a_{21}{b}_{12}+a_{22}{b}_{22}\end{array}\right]$ .

Calculation:

Matrices $0_{2\times 2}\times A=A\times 0_{2\times 2}=0_{2\times 2}.$

Zero matrix multiply by any number or matrix is zero.

  $\begin{array}{l}{0}_{2\times 2}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right].\\ A\times 0_{2\times 2}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right].\end{array}$

Hence, $0_{2\times 2}\times A=A\times 0_{2\times 2}=0_{2\times 2}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right].$