Phoebe

2021-02-06

Let the matrix A,B, C and D be as given. Find the product of the sum of A and B and the difference between C and D.
$A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],C=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right],D=\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$

Step 1
From the given matrices,
$A+B=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right]$
$C-D=\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]-\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 2\end{array}\right]$
Step 2
Now compute the product as follows.
$\left(A+B\right)\cdot \left(C-D\right)=\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right]\cdot \left[\begin{array}{cc}0& 0\\ 0& 2\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$
Thus, the product of the sum of A and B and the difference between C and D is zero matrix.

Jeffrey Jordon