Question Bank • QB_PARAMETRIC_EM_MAT_122NAT • 2 Marks
Section 1: Engineering Mathematics
Linear Algebra: Matrices & Determinants
Subtopic: Inverse of matrix
Question Statement:
Let $M$ be a $3 \times 3$ non-singular matrix with $\det(M) = 123$. The value of $\det(\operatorname{adj}(M))$ is ________ (answer in integer). (Variation QB_PARAMETRIC_EM_MAT_122)
Answer Key:
15129
Step-by-Step Solution & Explanation
For an $n \times n$ non-singular matrix $M$, the determinant of its classical adjugate matrix $\operatorname{adj}(M)$ is given by:
$\det(\operatorname{adj}(M)) = (\det(M))^{n-1}$
For order $n = 3$ and $\det(M) = 123$:
$\det(\operatorname{adj}(M)) = (123)^{3-1} = (123)^2 = 15129$
Therefore, the answer is $15129$.