Matrices & Determinants
Adjoint of a matrix
Grade Class 12
Question:
Let A = $\begin{bmatrix} l-3 & a & b \\ c & 6 & d \\ e & f & 9-l \end{bmatrix}$, B = adj(A) and C = adj(B). If |A| = 5, then tr(C) is (where |X|, tr(X) & adj(X) denote determinant value, trace and adjoint of matrix X respectively) -
(A) 5
(B) 12
(C) 30
(D) 60
Step-by-Step Solution
Key Concept: Use the property adj(adj(A)) = |A|^(n-2) * A, where n is the order of the matrix. Here n=3, so C = adj(adj(A)) = |A|^(3-2) * A = |A| * A. Thus, tr(C) = tr(|A| * A) = |A| * tr(A).
Given A is a 3x3 matrix. B = adj(A) and C = adj(B) = adj(adj(A)). We know that adj(adj(A)) = |A|^(n-2) * A. Since n=3, C = |A|^(3-2) * A = |A| * A. Given |A| = 5, so C = 5A. Therefore, tr(C) = tr(5A) = 5 * tr(A). The trace of A is (l-3) + 6 + (9-l) = l - 3 + 6 + 9 - l = 12. Thus, tr(C) = 5 * 12 = 60.
Correct Answer: 4