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:

You can proceed as follows:

Step-by-step explanation:

Suppose that the matrix $ABC$ is invertible, and suppose that at least one of the matrices $A,B,C$ is not invertible. Without loss of generality suppose that the matrix $A$ is not invertible.  Remember the important result that a matrix is invertible if and only if its determinant is nonzero. Then,

$\det (ABC)\neq 0.$

On the other hand, the determinant of a products of matrices is the product of the determinants of the matrices, that is to say,

$\det (ABC)=\det(A)\cdot \det(B)\cdot \det (C).$

But we supposed that $A$ is not invertible. Then $\det (A)=0$. Then $\det(A)\cdot \det(B)\cdot \det (C)=0$. This contradicts the fact that

$\det (ABC)=\det(A)\cdot \det(B)\cdot \det (C)$

and then the three matrices $A,B,\, \text{and}\, C$ must be invertible matrices.