Question

If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then the correct statement is:

• A. $A^2+5A-7I=O$
• B. $-A^2+5A+7I=O$
• C. $A^2-5A+7I=O$
• D. $A^2+5A+7I=O$

Answer 1

The correct option is C $A^2-5A+7I=O$

Now, $A^2=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$
$=\left[\begin{array}{cc}8& 5\\ -5& 3\end{array}\right]$
$\therefore A^2-5A+7I=\left[\begin{array}{cc}8& 5\\ -5& 3\end{array}\right]-\left[\begin{array}{cc}15& 5\\ -5& 10\end{array}\right]+\left[\begin{array}{cc}7& 0\\ 0& 7\end{array}\right]$
$=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$