Question

need help in linear algebra
here is the question

Select all statements below which are true for all invertible nxn matrices A & B

A. A^2 is invertible
B. (A+B)(A-B) = A^2-B^2
C. A+A^-1is invertible
D. ABA^-1 = B
E. (I - A)(I - A) =I -A^2
F. (AB)^-1 = A^-1B^-1

Thank you

Answer 1

I have trouble with understanding the theory

Answer 2

If A is invertible, then there exists another matrix A^(-1) (which we call the inverse of A) such that
\[A^{-1} A = I \]
where I is the identity matrix.

If A is invertible, do you think A^2 would be invertible?

Answer 3

ok i understand

Answer 4

yes a^2 is invertible

Answer 5

(A+B)(A-B) = A^2-B^2 this one is no right?

Answer 6

For the second one,
\[(A+B)(A-B) =A^2 -AB+BA-B^2\]
so
\[(A+B)(A-B) = A^2 - B^2\]
if and only if
\[AB = BA\]
if this is true, we say that A and B commute. Does this appear to have anything to do with invertibility?

Answer 7

because A^2-B^2 can't satisfy the equation right?

Answer 8

You are right. The fact that A and B are invertible does not mean that AB = BA.

Answer 9

ok got it and for the third one

Answer 10

Well, what do you think?

Answer 11

A+A^-1--> is not invertible right

Answer 12

since AA^-1 -> I

Answer 13

A+A^-1 have nothing to do with invertibility

Answer 14

If A is invertible, it does not necessarily mean that A + A^(-1) is also invertible. The only such examples I can think of use complex numbers, but that's okay.