det 1 2 1 2= 1*2 - 2*(1) = 2 -2 = 0
The above matrix has a zero determinant and is therefore singular. It has no inverse. It has two identical rows. In other words, the rows are not independent. If one row is a multiple of another, then they are not independent, and the determinant is zero. (Equivalently: If one column is a multiple of another, then they are not independent, and the determinant is zero.)
The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns). A square matrix An×n is non-singular only if its rank is equal to n.
What is the rank of the following matrix?
1 2 0 3 1 -2 3 0 0 0 4 8 2 4 0 6