Is the product of three non-square matrices possibly invertible if they produce a square matrix?

Question

My question is if it is possible that the product of 3 non-square matrices is invertible if this product is a square matrix. e.g.

$A=2\times3, B = 3 \times 4, C = 4 \times 2$

and

$D=ABC$ then $D= 2\times2$ Can D be invertible?

Thanks!

Daniel

Answer 1

What you get at the end is a square matrix.

So it can be invertible, as long as $\text{Det}\ D \neq 0$

Answer 2

It is necessary that each of the matrices has rank at least 2. They don't need to have full rank. Here is an example: $$ \begin{bmatrix} 1&0&0\\ 0&1&0\\ \end{bmatrix} \begin{bmatrix} 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0 \end{bmatrix} \begin{bmatrix} 1&0\\ 0&1\\ 0&0\\ 0&0 \end{bmatrix} = \begin{bmatrix} 1&0\\ 0&1\\ \end{bmatrix} $$

Why those matrices: we need each of the three matrices to be full rank, and each of those is the easiest example with the maximum number of linearly independent columns.