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Let $A$ be a square matrix of real numbers whose columns are (non-zero) orthogonal vectors.
a) Show that $A^{T} A$ is a diagonal matrix - whose inverse is thus obvious to compute.
b) Use this observation (or any other method) to discover a simple general formula for the inverse, $A^{-1}$ involving only its transpose, $A^{T}$, and $\left(A^{T} A\right)^{-1}$. In the special case where the columns of $A$ are orthonormal, your formula should reduce to $A^{-1}=A^{T}$.
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