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

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.