Question

Suppose that A and B are square matrices and that ABC is invertible. Show that each of A, B, and C is invertible.

Answer 1

Explanation:

Let A, B and C be square matrices, let
`D = ABC`. Suppose also that D is an invertible square matrix. Since D is an invertible matrix, then
`det (D) \\eq 0`. Now,
`det (D) = det (ABC) = det (A) det (B) det (C) \\eq 0`. Therefore,

`det (A) \\eq 0`

`det (B) \\eq 0`

`det (C) \\eq 0`

which proves that A, B and C are invertible square matrices.