Question

Let A and B be 4 x 4 matrices, with det(A) = 3 and det(B) = 5. Compute: 1. det(AB) – det(BA) 2. det(A²)+det(B2) 3. det(2A3B) 4. det(ATBA) 5. det(B-1AB)

Answer 1

We have

`\det(AB)=\det A\det B`

for any two matrices
`A,B`, and

`\det A^(-1)=\frac1{\det A}`

`\det A^\top=\det A`

`\det(kA)=k^n\det A`

where
`k` is a constant and
`n` is the size of the matrix
`A`.

1.
`\det(AB)-\det(BA)=\det A\det B-\det B\det A=0`

2.
`\det(A^2)+\det(B^2)=(\det A)^2+(\det B)^2=34`

3.
`\det(2A^3B)=2^4\det(A^3B)=16(\det A)^3\det B=2160`

4.
`\det(A^\top BA)=\det A^\top\det B\det A=45`

5.
`\det(B^(-1)AB)=\frac1{\det B}\det A\det B=\det A=3`