Elementary matrices are obtained from the identity matrix by row operations which will not change the character of the determinant. So since the identity matrix always has a non zero determinant, the elementary matrix also has a non zero determinant, and thereby invertible.

Why is elementary matrix invertible?

Every elementary matrix is invertible and its inverse is also an elementary matrix. In fact, the inverse of an elementary matrix is constructed by doing the reverse row operation on I. E−1 will be obtained by performing the row operation which would carry E back to I.

What makes a matrix invertible?

An invertible matrix is a square matrix that has an inverse. We say that a square matrix is invertible if and only if the determinant is not equal to zero. In other words, a 2 x 2 matrix is only invertible if the determinant of the matrix is not 0.