Question

Box all final answers. Little to no work will reccive little to no credit! I. (5 points) Let A and B be 4 x 4 matrices, with det(A) =-3 and det(B) = 2: Compute (a) det(AB) (b) det(B5) (c) det(2A) (d) det(ATA) (e) det(B-AB)

Answer 1

Answer: a) -6, b) 32, c) -48, d) 9, e) -12

Explanation:

Since we have given that

A and B are 4 × 4 matrices.

Here,

det (A) = -3

det (B) = 2

We need to find the respective parts:

a) det (AB)

`\mid AB\mid=\mid A\mid.\mid B\mid\\\\\mid AB\mid=-3* 2=-6`

b) det (B⁵ )

`\mid B^5\mid=\mid B\mid ^5=2^5=32`

c) det (2A)

Since we know that

`\mid kA\mid =k^n\mid A\mid`

so, it becomes,

`\mid 2A\mid =2^4\mid A\mid=16* -3=-48`

d)
`\bold{det(A^TA)}`

Since we know that

`\mid A^T\mid=\mid A\mid`

so, it becomes,

`\mid A^TA\mid=\mid A^T\mid * \mid A\mid=-3* -3=9`

e) det (B⁻¹AB)

As we know that

`\mid B^(-1)\mid =\mid B\mid`

so, it becomes,

`\mid B^(-1)AB}\mid =\mid B^(-1).\mid \mid A\mid.\mid B\mid=2* -3* 2=-12`

Hence, a) -6, b) 32, c) -48, d) 9, e) -12