If A= [ cc 1 & 4 1 & -3 ] and B= [ cc 1 & 2 -1 & -1 ] Find: i. (A… - Vidyadip

Question

If $A=\left[\begin{array}{cc}1 & 4 \\ 1 & -3\end{array}\right]$ and $B=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right]$ Find:
$i. (A+B)^2$
$ii. A^2+B^2$
$iii.$ Is $(A+B)^2=A^2+B^2$ ?

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Answer

$A+B=\left[\begin{array}{cc}1 & 4 \\ 1 & -3\end{array}\right]+\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right]$
$=\left[\begin{array}{cc}1+1 & 4+2 \\ 1-1 & -3-1\end{array}\right] =\left[\begin{array}{cc}2 & 6 \\ 0 & -4\end{array}\right]$
Now, $( A + B )^2=( A + B )( A + B )$
$\begin{array}{l}=\left[\begin{array}{cc}2 & 6 \\ 0 & -4\end{array}\right]\left[\begin{array}{cc}2 & 6 \\ 0 & -4\end{array}\right] \end{array} $
$=\left[\begin{array}{cc}(2)(2)+(6)(0) & (2)(6)+(6)(-4) \\ (0)(2)+(-4)(0) & (0)(6)+(-4)(-4)\end{array}\right] $
$ =\left[\begin{array}{cc}4 & -12 \\ 0 & 16\end{array}\right]$
$(ii)\ A^2=\left[\begin{array}{cc}1 & 4 \\ 1 & -3\end{array}\right]\left[\begin{array}{cc}1 & 4 \\ 1 & -3\end{array}\right] $
$=\left[\begin{array}{cc}1+4 & 4-12 \\ 1-3 & 4+9\end{array}\right] $
$ =\left[\begin{array}{cc}5 & -8 \\ -2 & 13\end{array}\right]$
$B ^2=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right]\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right]$
$\left[\begin{array}{cc}1-2 & 2-2 \\ -1+1 & -2+1\end{array}\right]$
$\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right] A^2+B^2=\left[\begin{array}{cc}5 & -8 \\ -2 & 13\end{array}\right]+\left[\begin{array}{cc}-1 & 0 \\ 0 & -1\end{array}\right]$
$A ^2+ B ^2=\left[\begin{array}{cc}4 & -8 \\ -2 & 12\end{array}\right]$
$(iii)$ No $,(A+B)^2 \neq A^2+B^2$.

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