Question

Characteristic equation for the matrix $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$ is

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

Answer 1

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

For characteristics equation : $f\left(\lambda \right)=|A-\lambda I|$
$\Rightarrow f\left(\lambda \right)=\left|\begin{array}{cc}1-\lambda & 2\\ 3& 4-\lambda \end{array}\right|\Rightarrow f\left(\lambda \right)=(1-\lambda )(4-\lambda )-6\Rightarrow f\left(\lambda \right)={\lambda }^2-5\lambda -2$
$\therefore f\left(A\right)=O\Rightarrow A^2-5A-2I=O$