Relationship Between Rank, Nullity, Determinant and Invertibility
Let
The following are all equivalent:
is invertible (bijective)
Using the same notation in the theorem above, we prove this using the following lemmas.
Proof
Assume that
Then since
On the converse, if
which is equivalent to
With
Proof
If
Similarly if
Proof
We can now prove the main result. (2) and (3) are equivalent from the rank nullity theorem. Since (2) implies injectivity, and (3) implies surjectivity, but (2) is equivalent to (3), each of these independently prove bijectivity, (1). The converse is also true, with (1) implying
Hence we have established
Then from matrix has non-zero determinant if and only if it is invertible we can conclude the final equivalenc.