Determinants and Invertiblity Theorem

Theorem

A matrix $M\in M_{n\times n}(\mathbb{R})$ is invertible $⟺$ $\det (M)\ne 0$

Proof

Proving $⟸$

1. We can prove the contrapositive first
2. Suppose $M$ is not invertible
3. In the REF of $M$, we have a row of zeroes
4. Therefore, one of the rows of M is a linear combination of other rows of $M$
5. Thus, the rows of $M$ are dependent, and we get $\det (M)=0$