Answer to Question #234618 in Linear Algebra for Bless

Question

Suppose that A, B, C are 3×3 matrices with det (A) = 2, det (B) = 3 and det (C) = 5. Compute the following determinants:

(a) det (AB)

(b) det (3AB^-2C²)

(c) det (A²C^TB^-1)

Answer 1

ANSWERS

$(a) \det (AB) = 6 \\ (b) \det (3AB^{-2}C^2) = 150 \\ (c) \det (A^2C^TB^{-1}) = \frac{20}{3}$

SOLUTIONS

$(a) \det(AB) = \det(A) \times \det(B)$

$= 2 \times 3$

$=6$

$(b) \det(3AB^{-2}C^2) = 3^3 \cdot \det(A) \cdot \det(B)^{-2} \cdot \det (C)^2$

$= 27 \times 2 \times \frac{1}{3^2} \times 5^2$

$= 27 \times 2 \times \frac {1}{9} \times 25$

$= 150$

$(c) \det(A^2C^TB^{-1}) = \det(A)^2 \cdot \det(C^T) \cdot \det (B^{-1})$

$= 2^2 \times 5 \times \frac {1}{3}$

$= 4 \times 5 \times \frac {1}{3}$

$= \frac {20}{3}$