Homework SolutionEigenvalue of a Matrix: Proof Involving Nonsingular Matrices

[chocolatefrog]

Proof involving nonsingular matrices.

Homework Statement

If (I + A) is nonsingular, prove that (I - A)(I + A)^-1 = (I + A)^-1(I - A), and hence (I - A)/(I + A) is defined for the matrix.

I've proved it like this:

Let (I - A)(I + A)^-1 = A, and (I + A)^-1(I - A) = B.
B^-1 = (I - A)^-1(I + A)
B^-1A = I
Premultiplying by B, we get A = B.

Is this proof correct?

 

[micromass]

Homework Statement

If (I + A) is nonsingular, prove that (I - A)(I + A)^-1 = (I + A)^-1(I - A), and hence (I - A)/(I + A) is defined for the matrix.

I've proved it like this:

Let (I - A)(I + A)^-1 = A, and (I + A)^-1(I - A) = B.
B^-1 = (I - A)^-1(I + A)
B^-1A = I
Premultiplying by B, we get A = B.

Is this proof correct?

You don't know that I-A is invertible. So (I-A)^-1 might not exist.

 

[chocolatefrog]

You don't know that I-A is invertible. So (I-A)^-1 might not exist.

Oh, I forgot to mention that A is known to be skew-symmetric. So, (I - A)^T = (I + A), which is nonsingular. And since a matrix is nonsingular iff its transpose is nonsingular, we could assume that (I - A)^-1 exists.

I can't seem to think beyond this point. If there's still an error somewhere in the proof, could you please point to it?