Question

If $A=\left[\begin{array}{cc}5& 3\\ -2& 1\end{array}\right],$ then $3A^2-20A+30I$ is equal to:

• A. $2A+3I$
• B. $2A-3I$
• C. $-2A-3I$
• D. $-2A+3I$

Answer 1

The correct option is C $-2A-3I$

The characteristics equation for $A$ is : $|A-\lambda I|=0$
$\Rightarrow \left|\begin{array}{cc}5-\lambda & 3\\ -2& 1-\lambda \end{array}\right|=0\Rightarrow (5-\lambda )(1-\lambda )+6=0\Rightarrow {\lambda }^2-6\lambda +11=0$
From Cayley-Hamilton formula : matrix $A$ will satisfy the characteristics equation,
$\Rightarrow A^2-6A+11I=O\Rightarrow 3A^2-18A+33I=O\Rightarrow 3A^2-20A+30I=-3I-2A$