Invertible matrix

Furthermore, the following properties hold for an invertible matrix A:

We can easily see the rank of this 2*2 matrix is one, which is n-1≠n, so it is a non-invertible matrix.

As an example of a non-invertible, or singular, matrix, consider the matrix

If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by

Matrices can also be inverted blockwise by using the following analytic inversion formula:

By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.

Decomposition techniques like LU decomposition are much faster than inversion, and various fast algorithms for special classes of linear systems have also been developed.