Question

Let $ A \in \mathbb{R} $ be a matrix of order 3x3 such that $$ \det(A) = -4 \quad \text{and} \quad A + I = \left[ \begin{array}{ccc} 1 & 1 & 1 \\2 & 0 & 1 \\4 & 1 & 2 \end{array} \right] $$ where $ I $ is the identity matrix of order 3. If $ \det( (A + I) \cdot \text{adj}(A + I)) $ is $ 2^m $, then $ m $ is equal to:

• A. 14
• B. 31
• C. 13
• D. 16

Hint:

The determinant of a matrix multiplied by its adjugate is the square of the determinant of the matrix.

Answer 1

The Correct Option is D

Given \( A + I \) and \( \det(A) = -4 \), first find the determinant of the matrix: \[ \det(A + I) = \det\left( \begin{array}{ccc} 1 & 1 & 1 2 & 0 & 1 4 & 1 & 2 \end{array} \right) \] Using cofactor expansion: \[ \det(A + I) = 16 \] Now, using the adjugate formula: \[ \det\left( (A + I) \cdot \text{adj}(A + I) \right) = \left( \det(A + I) \right)^2 = 16^2 = 2^{16} \] Thus, \( m = 16 \).