# Answer to Question #21511 in Algebra for lishan hailu

Question #21511

8) give matrices A and B of size 3x3. if det (A)=4, det (B)=-3, then det(A-1(2B)2)= ---------

Expert's answer

Denote by a^b the number a in power b, so for example 2^3= 2*2*2=8.

We will use the following properties of determinant:

1) Determinant of product of two matrices is equal to theproduct of their determinants

det(X*Y) = det(X) * det(Y)

In particular, det(X^2) = det(X) * det(X) = det(X)^2

2) Determinant of the inverse matrix is equal to1/determinant of initial matrix:

det(A^{-1}) = 1/det(A)

3) Determinant of matrix kX having size nxn is equal to det(kB) = k^n * det(B)

Thus

det(A^{-1}) * det((2B)^2) =

= 1/det(A) *(det(2B))^2 =

= 1/4 * (2^3 *det(B))^2 =

= 1/4 * (8 *(-3))^2 =

= 1/4 *(-24)^2 =

= 24/4 * 24 =

= 6 * 24 =

= 144

We will use the following properties of determinant:

1) Determinant of product of two matrices is equal to theproduct of their determinants

det(X*Y) = det(X) * det(Y)

In particular, det(X^2) = det(X) * det(X) = det(X)^2

2) Determinant of the inverse matrix is equal to1/determinant of initial matrix:

det(A^{-1}) = 1/det(A)

3) Determinant of matrix kX having size nxn is equal to det(kB) = k^n * det(B)

Thus

det(A^{-1}) * det((2B)^2) =

= 1/det(A) *(det(2B))^2 =

= 1/4 * (2^3 *det(B))^2 =

= 1/4 * (8 *(-3))^2 =

= 1/4 *(-24)^2 =

= 24/4 * 24 =

= 6 * 24 =

= 144

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