rank

Rank of a matrix

The rank of a matrix $A$ is the dimension of the column space of $A$, denoted $\text{rank}(A)$.
Row rank = col rank for any matrix $A\in K^{m\times n},\text{rank}(A)=\text{rank}(A^T)$.
So we can use both row and column operations to find the rank (unlike for finding the inverse or solving systems of equations, where we only use row operations).
$\text{rank}(A)\le \min \{m,n\}$ for any $A\in K^{m\times n}$.

The determinant of a square matrix $A\in K^{n\times n}$ is non-zero $⟺$ its rank is $n$.

$$\det A\ne 0⟺\text{rank}(A)=n$$

Proof:
Row operations don’t change the determinant. The row echelon form of a square matrix is upper triangular, whose determinant is the product of diagonal entries. Full rank means all $n$ pivots are non-zero, so $\det A\ne 0$. If rank $<n$, at least one pivot is zero, so $\det A=0$.