Question

If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$, then $A^2-5A$ is equal to

• A. $I$
• B. $-I$
• C. $7I$
• D. $-7I$

Answer 1

Answer: (D)

$-7I$

$A^2-5A=AA-5A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]-5\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$

$=\left[\begin{array}{cc}9-1& 3+2\\ -3-2& -1+4\end{array}\right]-\left[\begin{array}{cc}15& 5\\ -5& 10\end{array}\right]$

$=\left[\begin{array}{cc}8& 5\\ -5& 3\end{array}\right]-\left[\begin{array}{cc}15& 5\\ -5& 10\end{array}\right]$

$=\left[\begin{array}{cc}-7& 0\\ 0& -7\end{array}\right]=-7\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=-7I$