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Let A be a non-singular matrix. Show t...

Let `A` be a non-singular matrix. Show that `A^T A^(-1)` is symmetric if `A^2=(A^T)^2` .

Text Solution

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First let `A^T A^(−1)` be symmetric. Then
`(A^T A^(−1))^T=A^T A^(−1) `
⇒`(A^(−1))^T(A^T)^T=A^T A^(−1)`
⇒`(A^T)^(−1)A=A^T A^(−1) [∵(A^(−1))^T=(A^T)^(−1)]`
...
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