Question

If $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$, then $A^4-5A^3-A^2-4A-I$ is equal to

• A. zero matrix
• B. $I$
• C. $A$
• D. $A+I$

Answer 1

The correct option is D $A+I$

We know that every square matrix satisfies its characteristic equation.
$|A-\lambda I|=0$
$\left|\begin{array}{cc}1-\lambda & 2\\ 3& 4-\lambda \end{array}\right|=0$
$\Rightarrow {\lambda }^2-5\lambda -2=0$
$\Rightarrow {\lambda }^2=5\lambda +2$
$\Rightarrow A^2=5A+2I$
$\Rightarrow A^3=5A^2+2A$
$\Rightarrow A^4=5A^3+2A^2$
$\Rightarrow A^4-5A^3-A^2=A^2$
$\Rightarrow A^4-5A^3-A^2=5A+2I$
$\Rightarrow A^4-5A^3-A^2-4A-I=A+I$