Inverse of Inverse of Matrix

Theorem

Let $\mathbf A$ be a nonsingular matrix.

Then:

$\paren {\mathbf A^{-1} }^{-1} = \mathbf A$

That is, a nonsingular matrix equals the inverse of its inverse.


Proof

By definition of inverse matrix:

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

where $\mathbf I$ is the unit matrix.

Thus the inverse of $\mathbf A^{-1}$ is $\mathbf A$.

Hence the result.

$\blacksquare$

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