Calculate the determinant of the matrix Q=3 0 -2 1 4 5 -1 2 0.

Calculate the determinant of the matrix $Q = 31 - 1 042 - 2 50$ .

medium 3 marks Class 12 · Mathematics · Matrices

Answer

det (Q) = - 42

The correct answer is $det (Q) = - 42$ — Thus the determinant equals $- 42$ .

Use cofactor expansion along the first row: $det (Q) = 3 C_{11} + 0 C_{12} + (- 2) C_{13}$ .
Compute $C_{11} = det (4250) = 4 \cdot 0 - 5 \cdot 2 = - 10$ .
Compute $C_{13} = det (1 - 1 42) = 1 \cdot 2 - 4 (- 1) = 2 + 4 = 6$ ; the sign $(- 1)^{1 + 3} = + 1$ .
Substitute: $det (Q) = 3 (- 10) + (- 2) (6) = - 30 - 12 = - 42$ .
Thus the determinant equals $- 42$ .

Common mistake: Missing the sign change for the cofactor

C_{13}

or making arithmetic errors while evaluating

2 \times 6

Concept tested: Determinant of a $3\times3$ matrix