Let be an matrix over a field.
- Suppose is a matrix with columns , and let denote the th standard basis vector. Prove that is an inverse of if and only if is a solution of the matrix equation for each .
- Prove that has an inverse if and only if is row equivalent to the identity matrix.
- Prove that has an inverse if and only if and are row equivalent.
We begin with a lemma.
Lemma: Let and be matrices of dimension over a field. If , then . Proof: Thinking of and as linear transformations, since is injective, is injective. Since we are working over a vector space of finite dimension, is also surjective. So is an automorphism. Thus there exists a matrix such that , and we have . So .
Thus, in , to show a matrix is invertible it suffices to show left- or right-invertible.
- Suppose is an inverse of . Then we have . In particular, is a solution of the equation for each . Conversely, if for each , then . Thus is an inverse of .
- Suppose there is a matrix such that . In particular, is invertible, and so is row equivalent to the identity matrix. Conversely, if is row equivalent to the identity matrix, then for some invertible matrix . By the lemma, is an inverse of .
- Suppose is invertible with inverse . Then , so that and are row equivalent. Now suppose and are row equivalent, say by an invertible matrix . Then . Thus and , so that is an inverse of .