Question

Suppose that A and B are matrices of n X n. Show that if A is reversible, then then the `det⁡ (B) = det(A ^ (- 1) BA)` ?

Answer 1

This is a standard property of similar matrices. See below:

Explanation:

We know `det(AB) = det(A) det(B)`.

Also, since `AA^-1 = I`, we have
`det(A) det(A^-1) = det(AA^-1) = det(I) = 1`
`implies det(A^-1) = 1/det(A)`

So,

`det(A^-1BA) = det(A^-1)det(B)det(A) = 1/det(A) det(B) det(A) = det(B)`