The rank of a matrix can also be calculated using determinants. We can define rank using what interests us now.

The rank of a matrix is the order of the largest non-zero square submatrix.

See the following example.

1) Given , we eliminate rows or columns acording to the criterion to calculate the rank using the Gaussian elimination method. Thus,

Column can be discarded because all its elements are zero.

Column can be discarded because it is a linear combination of column and column . Specifically, .

2) Is there any non-zero square submatrix of order ?

Any non-zero element is a non-zero square submatrix, therefore we will look at those of higher order.

Is there any non-zero square submatrix of order ?

Yes, there is, therefore we will look for higher orders.

4) Is there any non-zero square submatrix of order ?

No, there is not. Therefore, rank , which is the order of the largest non-zero square submatrix.