Question

How is (A^-1)^T=(A^T)^T=A=(A^-1)^-1? And same with det(A)det(A^T) = det(A)det(A)=I

Answer 1

I don't see how (A^-1)^T = (A^-1)^-1

Answer 2

The inverse of a matrix then transposed is the identity matrix. This is an axiom. Try it out with a 2x2 matrix.
A transposed matrix multiplied by itself is the identity matrix. This is also an axiom. Try it with a 2x2 matrix.
The inverse equation, I am unsure about raising an inverse to an inverse.
det(A)*det(A) = I can be seen by using abstraction of a 2x2.
Hope this helps.
http://www.tutorsean.net

Answer 3

Oh nvm I see it b/c A^T = A^-1 so if you just change A to A^-1 it should still be the same

Answer 4

Just don't see how det(A)det(A^T) = det(A)det(A)

Answer 5

Originally I had A^TA=I

Answer 6

so then I took the det of both sides

Answer 7

Because det(A) = det(A^T)

Answer 8

but the determinant of I is det(A)det(A)?

Answer 9

which then equals 1

Answer 10

My Linear is a bit on the rusty side, and we did not cover proofs in my Linear Algebra class (regrets.), but I am now trying to learn the proofs of the axioms. You can check my website in the near future for the worked proof videos (free).

http://www.tutorsean.net

Answer 11

I will in sec. So how does det(A)det(A) = 1?

Answer 12

oh nvm I see if it's an identity okay nvm got it