Question

Let A be a 5x5 matrix, det(A)= -3. Find

a) det(-A)

b) det(A^5)

c) det(-2A^T)

d) det(A^-3)

Answer 1

Some properties of the determinant:

• For any two square matrices
  `A,B`, we have
  `\det(AB)=\det A\det B`.
• For an
  `n* n` matrix
  `A`,
  `\det(kA)=k^n\det A` for
  `k\in\Bbb R`.
• 
  `\det(A^\top)=\det A`
• For an invertible matrix
  `A`,
  `\det(A^(-1))=\frac1{\det A}`.

So

a.
`\det(-A)=(-1)^5\det A=3`

b.
`\det(A^5)=(\det A)^5=243`

c.
`\det(-2A^\top)=(-2)^5\det(A^\top)=(-2)^5\det A=96`

d. Since
`\det A\\eq0`, the inverse
`A^(-1)` exists, so
`A^(-n)=(A^(-1))^n`.

`\det(A^(-3))=\det((A^(-1))^3)=(\det(A^(-1)))^3=\frac1{(\det A)^3}=-\frac1{27}`