If A satisfies the equation x^3 -5x^2+4x+λ=0, then A^-1 exists if

Since A satisfies the equation

$x^3-5x^2+4x+λ=0$

$⇒A^3 -5A^2 + 4A + λI=0$

$⇒A (-A^2+5A-4 I) = λI$

$⇒A\left\{\frac{1}{λ}(-A^2+5A-4 I)\right\}=I$, if $λ≠0$

Hence, $A^{-1}$ exists and is equal to $\frac{1}{λ}(-A^2+5A-4 I)$ if $λ≠0$.