Suppose that a and b are square matrices and that abc is invertible. show that each of a, b, and c is invertible.

You can proceed as follows:

Step-by-step explanation:

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

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

But we supposed that is not invertible. Then . Then . This contradicts the fact that

and then the three matrices must be invertible matrices.

step-by-step explanation:

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