Question

If $A=\left[\begin{array}{cc}1& -2\\ 2& 3\end{array}\right],$ then the value of $A^3-4A^2+10A=$

• A. $A$
• B. $2A$
• C. $3A$
• D. $4A$

Answer 1

The correct option is C $3A$

The characteristic equation for $A$ is : $f\left(\lambda \right)=|A-\lambda I|$
$\Rightarrow f\left(\lambda \right)=\left|\begin{array}{cc}1-\lambda & -2\\ 2& 3-\lambda \end{array}\right|\Rightarrow f\left(\lambda \right)=(1-\lambda )(3-\lambda )+4\Rightarrow f\left(\lambda \right)={\lambda }^2-4\lambda +7$
From Cayley-Hamilton : matrix $A$ will satisfy the characteristic equation,
$\Rightarrow A^2-4A+7I=O\Rightarrow A^2-4A+10I=3I\text{pre-multiplying}A\text{both sides}\Rightarrow A^3-4A^2+10A=3A$