Let I be the identity matrix of order and for the matrix . Let B the inverse of the matrix . Then is equal to __________. [2025]

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Solution

Q.
Let $I$ be the identity matrix of order $3 \times 3$ and for the matrix $A = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}], | A | = - 1$ . Let B be the inverse of the matrix adj $(A a d j (A^{2}))$ . Then $| (λ B + I) |$ is equal to __________. [2025]

Ans.
(38)

Given, $A = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}]$

$\Rightarrow | A | = [\begin{matrix} λ & 2 & 3 \\ 4 & 5 & 6 \\ 7 & - 1 & 2 \end{matrix}] = - 1$

$\Rightarrow λ (16) - 2 (- 34) + 3 (- 39) = - 1$

$\Rightarrow 16 λ = 48 \Rightarrow λ = 3$

Given, $B^{- 1} = adj (A \cdot adj (A^{2}))$

Let $C = A \cdot adj (A^{2})$

$A C = A^{2} adj (A^{2}) = | A |^{2} \cdot I = I$

$\Rightarrow C = A^{- 1}$ [ $∵$ |A| = – 1]

Now, $B^{- 1} = adj (A^{- 1}) \Rightarrow B = adj (A)$

Now, $λ B + I = 3 B + I$

Let P = 3B + $I$

$\Rightarrow$ P = 3 adj(A) + $I$

$\Rightarrow$ AP = A 3 adj(A)+ (A)

$\Rightarrow$ AP = 3 $| A |$ $\cdot$ $I$ + A $\Rightarrow$ AP = A – 3 $I$

$\Rightarrow$ $| A P |$ $=$ $| A - 3 I |$

$| A |$ $\cdot$ $| P |$ $=$ $[\begin{matrix} 0 & 2 & 3 \\ 4 & 2 & 6 \\ 7 & - 1 & - 1 \end{matrix}]$

= 0 – 2(–46) + 3(–18)

= 92 – 54 = 38

$\Rightarrow$ $| P |$ = 38.

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