Chapter 16.3, Problem 20WE

To determine

To find: the $I_{2\times 2}.C$ and $C{I}_{2\times 2}$ using given matrices.

Answer to Problem 20WE

  $I_{2\times 2}\times C=A\times I_{2\times 2}=C=\left[\begin{array}{cc}-2& 0\\ 0& 1\end{array}\right].$

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:

The identity property means that anything multiplied by an identity matrix is itself.

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:

The identity property means that anything multiplied by an identity matrix is itself.

Matrices $I_{2\times 2}\times C=A\times I_{2\times 2}=C.$

  $\begin{array}{l}C=\left[\begin{array}{cc}-2& 0\\ 0& 1\end{array}\right].\\ I\times C=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\times \left[\begin{array}{cc}-2& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}-2& 0\\ 0& 1\end{array}\right].\\ I\times C=C.\end{array}$

Hence, $I_{2\times 2}\times C=A\times I_{2\times 2}=C=\left[\begin{array}{cc}-2& 0\\ 0& 1\end{array}\right].$