Question

Let I be the identity matrix of order 3 × 3 and for the matrix $ A = \begin{pmatrix} \lambda & 2 & 3 \\ 4 & 5 & 6 \\ 7 & -1 & 2 \end{pmatrix} $, $ |A| = -1 $. Let B be the inverse of the matrix $ \text{adj}(A \cdot \text{adj}(A^2)) $. Then $ |(\lambda B + I)| $ is equal to _______

Hint:

Use the properties of adjoint and inverse of matrices, such as \( A \cdot \text{adj}(A) = |A| I \) and \( \text{adj}(A^{-1}) = (\text{adj}(A))^{-1} \), to simplify the expression for \( B \). Then, use the properties of determinants, such as \( |AB| = |A| |B| \), to find the value of \( |\lambda B + I| \). Be careful with matrix operations and determinant calculations.

Answer 1

Correct Answer: 38

First, we find the value of \( \lambda \) using \( |A| = -1 \): \[ |A| = \begin{vmatrix} \lambda & 2 & 3 4 & 5 & 6 7 & -1 & 2 \end{vmatrix} = \lambda(10 - (-6)) - 2(8 - 42) + 3(-4 - 35) = -1 \] \[ 16\lambda - 2(-34) + 3(-39) = -1 \] \[ 16\lambda + 68 - 117 = -1 \] \[ 16\lambda - 49 = -1 \implies 16\lambda = 48 \implies \lambda = 3 \] Given \( B^{-1} = \text{adj}(A \cdot \text{adj}(A^2)) \). Let \( C = A \cdot \text{adj}(A^2) \). We know \( A^2 \cdot \text{adj}(A^2) = |A^2| I = |A|^2 I = (-1)^2 I = I \). So, \( \text{adj}(A^2) = (A^2)^{-1} \). 
Then \( C = A (A^2)^{-1} = A A^{-2} = A^{-1} \). Thus, \( B^{-1} = \text{adj}(A^{-1}) \). 
Using the property \( \text{adj}(M^{-1}) = (\text{adj}(M))^{-1} \), we have \( B^{-1} = (\text{adj}(A))^{-1} \). 
Therefore, \( B = \text{adj}(A) \). We need to find \( |\lambda B + I| = |3 \text{adj}(A) + I| \). 
Let \( P = 3 \text{adj}(A) + I \). Then \( AP = A(3 \text{adj}(A) + I) = 3 A \text{adj}(A) + A = 3 |A| I + A = 3(-1) I + A = A - 3I \). \[ |AP| = |A - 3I| \] \[ |A| |P| = \begin{vmatrix} 3 - 3 & 2 & 3 4 & 5 - 3 & 6 7 & -1 & 2 - 3 \end{vmatrix} = \begin{vmatrix} 0 & 2 & 3 \\ 4 & 2 & 6 \\ 7 & -1 & -1 \end{vmatrix} \] \[ |A| |P| = 0(..) - 2(4(-1) - 6(7)) + 3(4(-1) - 2(7)) = -2(-4 - 42) + 3(-4 - 14) = -2(-46) + 3(-18) = 92 - 54 = 38 \] Since \( |A| = -1 \), we have \( (-1) |P| = 38 \), so \( |P| = -38 \). 
Therefore, \( |\lambda B + I| = |P| = |-38| = 38 \).