det 1 2 1 2 = 1*2 - 2*(1) = 2 -2 = 0
The above matrix has a zero determinant and is therefor singular. It has no inverse. The matrix has two identical rows (and two identical columns). 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 col 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