Can the cross product of two non-invertible matrices be invertible?

Question

To put it better, if A and B are non-invertible matrices (for whatever reason), can the matrix AB be invertible?

Just used to help understand a Linear Transformation assignment question, don't necessarily need an explanation as to why, but it would help. Thankyou.

By $AB$ do you just mean the usual product of matrices? I've never seen this called the "cross product". Also, are $A$ and $B$ supposed to be square? — Eric Wofsey, Feb 03 '16 at 09:08
Yes sorry, cross product was what it was called in the high school I went to, and it's been stuck in my head. And no, in this case A is 2x3, and B is 3x2. — Aidan Leith, Feb 03 '16 at 09:16
Does this answer your question? If the product of two square matrices is invertible, then both matrices are invertible — , Feb 10 '21 at 07:53

score 3 · Answer 1 · edited Jun 12 '20 at 10:38

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$A,B$ square: never See for example How to prove and interpret rank(AB)$\le$ min(rank(A),rank(B))? But... $$\pmatrix{1&0}\pmatrix{1\cr0} = (1).$$

EDIT: more general and abstract example. For $\dim V = n$, take $$A: V\times V\longrightarrow V,\qquad A(x,y) = x,$$ $$B: V\longrightarrow V\times V,\qquad B(x) = (x,0),$$ and $AB =$ identity.

edited Jun 12 '20 at 10:38

Community

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answered Feb 03 '16 at 09:11

Martín-Blas Pérez Pinilla

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can you giver higher dimension example or generally for dimension of AB being n*n – mahdi Mar 28 '18 at 05:32
@mahdi, see the edit. – Martín-Blas Pérez Pinilla Mar 28 '18 at 07:54

score 1 · Answer 2 · answered Feb 03 '16 at 09:33

1) A square matrix is non-invertible if its determinant vanishes.

2) The determinant of a matrix product of two square matrices is the product of the determinants of the two individual matrices.

So, for square matrices the product is not invertible if one of the factors is not. For non-square matrices (which are non-invertible) I do not know if in some cases an invertible product can occur.

Can the cross product of two non-invertible matrices be invertible?

2 Answers2