Question

If $A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$ and $I$ is the unit matrix of order $2$ , then $A^2$ equals

• A. $4A-3I$
• B. $3A-4I$
• C. $A-I$
• D. $A+I$

Answer 1

Answer: (A)

$A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$
$A^2=A\times A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]\times \left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$
$=\left[\begin{array}{cc}4+1& -2-2\\ -2-2& 1+4\end{array}\right]=\left[\begin{array}{cc}5& -4\\ -4& 5\end{array}\right]$
From the given option to get first entry i.e $5$ of $A^2$,
only option (A) is satisfied. Or we can check as below
$4A-3I=\left[\begin{array}{cc}8& -4\\ -4& 8\end{array}\right]-3\times \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}8& -4\\ -4& 8\end{array}\right]-\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
$=\left[\begin{array}{cc}5& -4\\ -4& 5\end{array}\right]=A^2$