Question

If $A=\left[\begin{array}{cc}2& 1\\ -1& 2\end{array}\right],B=\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right],C=\left[\begin{array}{cc}1& -3\\ 2& 1\end{array}\right]$, then which of the following is true

• A. $A+B=B+A$ and $A+(B+C)=(A+B)+C$
• B. $A+B=B+A$ and $AC=BC$
• C. $A+B=B+A$ and $AB=BC$
• D. $AC=BC$ and $A=BC$

Answer 1

Answer: (A)

$A+B=B+A$ and $A+(B+C)=(A+B)+C$

Consider, $A+B=\left[\begin{array}{cc}2& 1\\ -1& 2\end{array}\right]+\left[\begin{array}{cc}1& -2\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}3& -1\\ 1& 3\end{array}\right]$

And $C+B+A=\left[\begin{array}{cc}3& -1\\ 1& 3\end{array}\right]+\left[\begin{array}{cc}1& -3\\ 2& 1\end{array}\right]=\left[\begin{array}{cc}4& -4\\ 3& 4\end{array}\right]$

By associative property and commutativity property of matrices of addition, we have

$A+B=B+A$ and $A+B+C=A+(B+C)=(A+B)+C$