Invertibility of a matrix

By determinant

Let $A$ be an $n\times n$ matrix of real or complex numbers. Then the inverse of A exists if and only if $\det A\ne 0$.

By row reduction

A matrix $A$ is invertible $⟺$ it can be row reduced into identity matrix.

Tip

If a matrix $A$ cannot be row reduced into $I_n$ then it means that there are rows of zeroes in the reduced system, implying determinant of reduced system is $0$. Knowing that elementary row operations do not create/remove zero determinant, this means that $\det A$ was actually $0⟹$ $A$ is not invertible.