# Is the inverse of a matrix also invertible?

Nov 5, 2015

Only if its determinant is non-zero

#### Explanation:

${A}^{- 1}$ exists $\iff \det \left(A\right) \ne 0$

Nov 5, 2015

Yes.

#### Explanation:

If a matrix $A$ has an inverse ${A}^{- 1}$ then it satisfies:

$A {A}^{- 1} = {A}^{- 1} A = I$

Looking at this a different way, the matrix $A$ satisfies the definition of an inverse for ${A}^{- 1}$, so ${A}^{- 1}$ is also invertible (with inverse $A$).