Rank Theorem

Definition

$\text{col-rank}(\mathbf{M})=\text{row-rank}(\mathbf{M})$

 The Column Rank and the Row Rank are always equal

Proof

Let a matrix $A$ be the echelon form of the matrix $M$. Then, $\text{col-rank}(\mathbf{M})=\text{col-rank}(\mathbf{A})$ and $\text{row-rank}(\mathbf{M})=\text{row-rank}(\mathbf{A})$ by the property of determinant

Since the number of columns or rows which containing pivots are the column or row rank of $A$. $\text{col-rank}(\mathbf{A})=\text{row-rank}(\mathbf{A})$

$\therefore \text{col-rank}(\mathbf{M}) = \text{row-rank}(\mathbf{M})$$