If matrix A is invertible then so it it transpose A^t

**Craka** · April 2nd 2009, 03:39 PM

How do I prove or show this? Not sure how to go about it.

**o_O** · April 2nd 2009, 04:13 PM

If you can prove that both: $A^T \left(A^{-1}\right)^T = \left(A^{-1}\right)^T A^T = I$

then by definition, we've shown that $A^T$ is invertible.

So for: $A^T \left(A^{-1}\right)^T = I$

Use the fact that: $X^TY^T = (YX)^T$

So we see that: $A^T\left(A^{-1}\right)^T = \left(A^{-1}A\right)^T = I^T = I$

Now finish off by proving the other.

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If matrix A is invertible then so it it transpose A^t

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