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

The correct option is A $4A-3I$

Now $A^2=A.A=\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]$
$=\left[\begin{array}{cc}4+1& -2-2\\ -2-2& 1+4\end{array}\right]$
$=\left[\begin{array}{cc}5& -4\\ -4& 5\end{array}\right]$
and $4A-3I=4\left[\begin{array}{cc}2& -1\\ -1& 2\end{array}\right]-3\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$=\left[\begin{array}{cc}8& -4\\ -4& 8\end{array}\right]-\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$
$=\left[\begin{array}{cc}5& -4\\ -4& 5\end{array}\right]$
$\therefore A^2=4A-3I$