So, for example:
Put A in row echelon form to get:
Now, since has two nonzero rows, you can conclude that .
However, once you know the nullspace of a matrix, you usually don't need to go through the above process to find the rank.
To find the nullspace of A, simply solve the system .
For example, using the same matrix:
Augment this matrix with the 0 column vector and reduce:
The dimension of the nullspace is called the nullity. Knowing the nullity of a matrix allows you to find the rank very easily, without finding the row or column space: For any matrix, .
For our matrix, we see that , which agrees with what we found earlier.
I hope that helped!