Applications of determinant

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Question

$If A is a non-zero matrix such that A^{2} = A, satisfving (I - λ A)^{- 1} = I - 3 A, where I is unit matrix of same$
$order as that of A, then the value of λ, where | I - λ A | \neq 0 is$

Moderate

Solution

$\begin{aligned} (I - λ A)^{- 1} = (I - 5 A) \\ ∴ I = (I - 3 A) (I - λ A) \end{aligned}$
$\begin{array}{l} \Rightarrow I = I - λ A - 3 A + 3 λ A^{2} \\ \Rightarrow O = - λ A - 3 A + 3 λ A \\ \Rightarrow (2 λ - 3) A = O \\ \Rightarrow 2 λ - 3 = 0 \\ \Rightarrow λ = 1.5 \end{array}$

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