Question

Let $A=\left[\begin{array}{cc}2& 1\\ 0& -2\end{array}\right],B=\left[\begin{array}{cc}4& 1\\ -3& -2\end{array}\right]$ and $C=\left[\begin{array}{cc}-3& 2\\ -1& 4\end{array}\right]$
Find: $A^2+AC-5B$

• A. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
• B. $\left[\begin{array}{cc}23& 3\\ 17& 6\end{array}\right]$
• C. $\left[\begin{array}{cc}-23& 3\\ 17& -6\end{array}\right]$
• D. $\left[\begin{array}{cc}-23& 3\\ 17& 6\end{array}\right]$

Answer 1

The correct option is D $\left[\begin{array}{cc}-23& 3\\ 17& 6\end{array}\right]$

$A^2=\left[\begin{array}{cc}2& 1\\ 0& -2\end{array}\right]\left[\begin{array}{cc}2& 1\\ 0& -2\end{array}\right]\left[\begin{array}{cc}2\times 2+1\times 0& 2\times 1+1\times -2\\ 0\times 2+-2\times 0& 0\times 1+(-2)(-2)\end{array}\right]$
$=\left[\begin{array}{cc}4& 0\\ 0& 4\end{array}\right]$
$AC=\left[\begin{array}{cc}2& 1\\ 0& -2\end{array}\right]\left[\begin{array}{cc}-3& 2\\ -1& 4\end{array}\right]$
$=\left[\begin{array}{cc}-3\times 2+1\times -1& 2\times 2+1\times 4\\ 0\times -3+(-2)\times (-1)& 0\times 2+(-2\times 4)\end{array}\right]$
$=\left[\begin{array}{cc}-7& 8\\ 2& -8\end{array}\right]$
$5B=\left[\begin{array}{cc}20& 5\\ -15& -10\end{array}\right]$
$A^2+AC-5B=\left[\begin{array}{cc}4& 0\\ 0& +4\end{array}\right]+\left[\begin{array}{cc}-7& 8\\ 2& -8\end{array}\right]-\left[\begin{array}{cc}20& 5\\ -15& -10\end{array}\right]$
$=\left[\begin{array}{cc}-23& 3\\ 17& 6\end{array}\right]$