A matrix A of size 4 × 4 has det(A) = 6. What is det(3A⁻¹)? Give the answer as a fraction in lowest terms.

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A matrix $A$ of size $4 \times 4$ has $det (A) = 6$ . What is $det (3 A^{- 1})$ ? Give the answer as a fraction in lowest terms.

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Hint

Pulling a scalar out of a matrix costs one factor per row, and inverting a matrix inverts its determinant.

Solution

Multiplying a $4 \times 4$ matrix by $3$ multiplies each of its $4$ rows by $3$ , so the determinant gains the factor $3^{4} = 81$ .
Because $det (A) \times det (A^{- 1}) = det (I) = 1$ , we have $det (A^{- 1}) = \frac{1}{6}$ .
So $det (3 A^{- 1}) = 81 \times \frac{1}{6} = \frac{27}{2}$ .

Answer: $\frac{27}{2}$