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 be an matrix. Then the following are equivalent:

  • 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