I have been introduced to a method for calculating the rank of and $m\times n$ matrix using determinants. The method states that the rank of our matrix is equal to the order of the largest square submatrix that has non zero determinant. My problem with the method is that a matrix has too many submatrices, for example my professor asked me to evaluate the rank of a specific $5\times 5$ matrix,(I know from the solution that rank of the specific matrix is 3. But if I were to solve the problem using the method I have been introduced I would have to verify that the determinant of each of the $4\times 4$ submatrices is zero. But there are 25 such matrices hence the methis seems inefficient to me. So I am wondering are there any tricks to the method i.e is there a way to come to the conclusion without having to check the determinant for each $4\times 4$ matrix?
Thanks for any help