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

Answer:

Step-by-step 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 $det (D) \neq 0$. Now, $det (D) = det (ABC) = det (A) det (B) det (C) \neq 0$. Therefore,

$det (A) \neq 0$

$det (B) \neq 0$

$det (C) \neq 0$

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