det 1 2 1 2= 1*2 - 2*(1) = 2 -2 = 0

The matrix has a zero determinant and is therefor singular. It has no inverse. If you look the matrix you see that it 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 **A**_{n×n} is non-singular only if its
rank is equal to n.