Not all matrices have inverse, and we call non-invertible matrices singular.
One way to calculate matrix inverse is by the Gauss–Jordan elimination.
Equivalent Conditions
There are a lot of equivalent conditions for invertible:
Let
is invertible (see determinant) (see nullspace (kernel)) - If
is a column vector in , there is a unique column vector in satisfying - All the rows and columns of
are linearly independent - The transpose
of is invertible - All of the eigenvalues of
are non-zero - Its rank is equal to
Some Theorems about Invertible Matrices
- If
exists, then it is unique - If
and exists, then - If
exists, then