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squaremeplz
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Homework Statement
Prove that if A is nonsingular then A^T is nonsingular and
(A^T)^(-1) = (A^(-1))^T
Homework Equations
(AB)^T = (B^T)(A^T)
The Attempt at a Solution
Step 1: Multiply both sides by B^T
B^T * (A^T)^(-1) = B^T * (A^(-1))^T
B^T * (A^T)^(-1) = (A^(-1)*B)^T
(A^(-1)*B)^T = (A^(-1)*B)^T
I feel that my last step is wrong. Any suggestions would be helpful.
I was also noticing that if I take just the right side of the equation and do this
A^T*(A^(-1))^T = (A^(-1)*A)^T
= I^T = I
which suggests that the left side has to equal I if I multiply it by A^T as well so
A^T*(A^T)^(-1) = I
which makes sense just looking at it.
Thanks!
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