Question

If $A$ satisfies the equation $x^3-5x^2+4x+\lambda =0$, then $A^{-1}$ exists if

• A. $\lambda \ne 1$
• B. $\lambda \ne 2$
• C. $\lambda \ne -1$
• D. $\lambda \ne 0$

Answer 1

Answer: (D)

$\lambda \ne 0$

Since, A satisfies the equation
$x^3-5x^2+4x+\lambda =0$
$\Rightarrow A^3-5A^2+4A+\lambda =O$
$\Rightarrow A^3{A}^{-1}-5A^2{A}^{-1}+4A{A}^{-1}+\lambda A^{-1}=O$
$\Rightarrow A^2-5A+4I+\lambda A^{-1}=O$
So, $A^{-1}$ exists if $\lambda \ne 0$