Rank of a square matrix
is found by finding the determinant. If the determinant is non-zero, the dimension of the square matrix is the rank.
If the determinant of A of size n x n is zero, then it is possible that there are two identical rows or columns. Or, algebraic sum of some rows or columns is equal to a row or column. Then the rank cannot be n.
We remove one of the identical rows or columns and then find the determinant of n-1 x n-1 matrix. If its determinant is non zero, then n-1 will be the rank of the matrix A.
Determinant is found by the usual method.
Determinant is zero because 3nd row is a multiple of 2nd row.
Let us remove the third row and column. and find the determinant again of the 2x2 matrix.