If det C = 12 for a 3×3 matrix C, what is det (5 C^-1)?

If $det C = 12$ for a $3 \times 3$ matrix $C$ , what is $det (5 C^{- 1})$ ?

medium MCQ 2 marks Class 12 · Mathematics · Determinants

Answer

\frac{125}{12}

The correct answer is $\frac{125}{12}$ — Compute $5^{3} = 125$ and multiply: $125 \times \frac{1}{12} = \frac{125}{12}$ .

For any invertible matrix $C$ , $det (C^{- 1}) = \frac{1}{d e t C} = \frac{1}{12}$ .
Multiplying a $3 \times 3$ matrix by a scalar $k$ multiplies its determinant by $k^{3}$ . Here $k = 5$ , so $det (5 C^{- 1}) = 5^{3} det (C^{- 1})$ .
Compute $5^{3} = 125$ and multiply: $125 \times \frac{1}{12} = \frac{125}{12}$ .

Common mistake: Using

5

instead of

5^{3}

when scaling a

3 \times 3

matrix.

Concept tested: Determinant of scalar multiple of inverse matrix