# Quick Linear Algebra Proof

#### squaremeplz

1. Homework Statement

Prove that if A is nonsingular then A^T is nonsingular and

(A^T)^(-1) = (A^(-1))^T

2. Homework Equations

(AB)^T = (B^T)(A^T)

3. 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!

Last edited:
Related Calculus and Beyond Homework Help News on Phys.org

#### Dick

Science Advisor
Homework Helper
Start from A*A^(-1)=I. Now take the transpose of both sides and use your relevant equation. Stare at it for a while and figure out what it means.

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