# What is the rank of a matrix?

Jul 24, 2018

#### Explanation:

Let $A$ be a $\left(m \times n\right)$ matrix.

Then $A$ consists of $n$ column vectors $\left({a}_{1} , {a}_{2} , \ldots {a}_{n}\right)$ which are $m$ vectors.

The rank of $A$ is the maximum number of linearly independent column vectors in $A$, that is, the maximun number of independent vectors among $\left({a}_{1} , {a}_{2} , \ldots {a}_{n}\right)$

If $A = 0$, the rank of $A$ is $= 0$

We write $r k \left(A\right)$ for the rank of $A$

To find the rank of a matrix $A$, use Gauss elimination.

The rank of the transpose of $A$ is the same as the rank of $A$.

$r k \left({A}^{T}\right) = r k \left(A\right)$