# Invertible Dimensional Matrices

#### eleventhhour

Suppose A, B, and C are invertible 4x4 dimensional matrices with the properties that det(A)=3, det(B)=5, and det(C)=2. Calculate the determinant of:

(3A^(-1)BCC^(T)A^(3))

I understand matrices and determinants but I can't figure out how to start this question...Could someone help? Once I know how to set it up I think I'd be able to solve it.
Thank you so much.

#### romsek

MHF Helper
Suppose A, B, and C are invertible 4x4 dimensional matrices with the properties that det(A)=3, det(B)=5, and det(C)=2. Calculate the determinant of:

(3A^(-1)BCC^(T)A^(3))

I understand matrices and determinants but I can't figure out how to start this question...Could someone help? Once I know how to set it up I think I'd be able to solve it.
Thank you so much.
$3A^{-1} B C C^T A^3$

Some properties of determinants assuming square and invertible matrices $A, B$

$det(AB) = det(A) det(B)$

$det(A^T)=det(A)$

$det\left(A^{-1}\right) = \dfrac {1}{det(A)}$

$det\left(A^k\right) = \det(A)^k$

so using these properties we get

$det \left(3A^{-1} B C C^T A^3\right) = \\ \\ 3 \dfrac{1}{det(A)} det(B) det(C) det(C) det(A)^3 = \\ \\ (3) \left(\dfrac 1 3 \right)(5)(2)(2)(3^3) =540$