217k views
1 vote
Suppose that A and B are square matrices and that ABC is invertible. Show that each of A, B, and C is invertible.

1 Answer

4 votes

Answer:

You can proceed as follows:

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)\\eq 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.

User Aviram
by
8.4k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories