Question

Select all statements below which are true for all invertible n×n matrices A and B A. AB=BA
B. (A+B)(A−B)=A2−B2
C. A+A−1 is invertible
D. A3B8 is invertible
E. (ABA−1)4=AB4A−1
F. (A+A−1)9=A9+A−9

Answer 1

where are stuck...what have you tried?

Answer 2

I have computed each statement, but I am still not getting the right combination

Answer 3

have you figured out the answer to any of these?

Answer 4

no, I do not understand the rules of invertible matrices

Answer 5

do you remember the rules of (square) matrix multiplication...in particular that matrix multiplication is not commutative?

Answer 6

also your notation is crazy..I believe I have figured out what you have typed...for example is

(A+A−1)9=A9+A−9

\[(A+A^{-1})^9=A^9-A^{-9}\]

Answer 7

Yes sorry I am new to this website

Answer 8

C and D is always right. A and B are equivalent but not always correct, neither does F. As for E, i am sorry but i dont understand ur notation.

Answer 9

If \(A=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\\ \end{array}\right]\) then \(A^{-1}=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\\ \end{array}\right]\)

and thus \(A+A^{-1}=\left[\begin{array}{cc}0 & 0 \\ 0 & 0\\ \end{array}\right]\) which is not invertable