• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Quick Linear Algebra Proof

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:

Dick

Science Advisor
Homework Helper
26,258
618
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.
 

Related Threads for: Quick Linear Algebra Proof

Replies
2
Views
1K
  • Posted
Replies
3
Views
972
  • Posted
Replies
7
Views
2K
Replies
1
Views
728
Replies
5
Views
676

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top