Question
Let $A=\left[\begin{array}{ll}4 & -2 \\ 6 & -3\end{array}\right], B=\left[\begin{array}{cc}0 & 2 \\ 1 & -1\end{array}\right]$ and $C=\left[\begin{array}{cc}-2 & 3 \\ 1 & -1\end{array}\right]$ Find $A^2+A+B C$
✓
Answer
$A^2=\left[\begin{array}{cc}4 & 2 \\ 6 & -3\end{array}\right]\left[\begin{array}{cc}4 & -2 \\ 6 & -3\end{array}\right]$
$=\left[\begin{array}{ll}16-12 & -8+6 \\ 24-18 & -12+9\end{array}\right]$
$=\left[\begin{array}{ll}4 & -2 \\ 6 & -3\end{array}\right]$
$B C=\left[\begin{array}{cc}0 & 2 \\ 1 & -1\end{array}\right]\left[\begin{array}{cc}-2 & 3 \\ 1 & -1\end{array}\right]$
$=\left[\begin{array}{cc}0+2 & 0-2 \\ -2-1 & 3=1\end{array}\right]$
$=\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
$A^2-A=B C=\left[\begin{array}{ll}4 & -2 \\ 6 & -3\end{array}\right]-\left[\begin{array}{cc}4 & -2 \\ 6 & -3\end{array}\right]+\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
$=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]+\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
$=\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
$=\left[\begin{array}{ll}16-12 & -8+6 \\ 24-18 & -12+9\end{array}\right]$
$=\left[\begin{array}{ll}4 & -2 \\ 6 & -3\end{array}\right]$
$B C=\left[\begin{array}{cc}0 & 2 \\ 1 & -1\end{array}\right]\left[\begin{array}{cc}-2 & 3 \\ 1 & -1\end{array}\right]$
$=\left[\begin{array}{cc}0+2 & 0-2 \\ -2-1 & 3=1\end{array}\right]$
$=\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
$A^2-A=B C=\left[\begin{array}{ll}4 & -2 \\ 6 & -3\end{array}\right]-\left[\begin{array}{cc}4 & -2 \\ 6 & -3\end{array}\right]+\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
$=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]+\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
$=\left[\begin{array}{cc}2 & -2 \\ -3 & 4\end{array}\right]$
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