Inverse of Inverse of Matrix
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Theorem
Let $\mathbf A$ be an invertible matrix.
Then:
- $\paren {\mathbf A^{-1} }^{-1} = \mathbf A$
That is, an invertible 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.
$\blacksquare$