Janelle S. answered 09/24/20
Penn State Grad for ME, Math & Test Prep Tutoring (10+ yrs experience)
1) ABA-1 = B => FALSE
Multiply both sides by A: ABA-1A = BA => AB = BA
Matrix multiplication is not commutative (you cannot switch the order of the factors), so AB does not equal BA.
2) (A + A-1)9 = A9 A-9 => FALSE
Using binomial expansion:
(A + A-1)9 = A9 + 9(A)8(A-1)1 + 36(A)7(A-1)2 + 84(A)6(A-1)3 + 126(A)5(A-1)4 + 126(A)4(A-1)5 + 84(A)3(A-1)6 + 36(A)2(A-1)7 + 9(A)1(A-1)8 + (A-1)9
To simplify, use the rules (Am)n = Amn and AmAn = Am+n:
= A9 + 9(A)8-1 + 36(A)7-2 + 84(A)6-3 + 126(A)5-4 + 126(A)4-5 + 84(A)3-6 + 36(A)2-7 + 9(A)1-8 + A-9
= A9 + 9(A)7 + 36(A)5 + 84(A)3 + 126A + 126(A)-1 + 84(A)-3 + 36(A)-5 + 9(A)-7 + A-9
3) A + B is invertible => TRUE
The sum of two invertible matrices is always invertible.
4) (In - A) (In + A) = In - A2 => TRUE
(In - A) (In + A) = In2 + InA - AIn - A2 = In + A - A - A2 = In - A2
Since In is the identity matrix, In2 = In, AIn = A, and InA = A.
5) (A + B)2 = A2 + B2 + 2AB => FALSE
(A + B)2 = (A + B) (A + B) = A2 + AB + BA + B2
Matrix multiplication is not commutative (you cannot switch the order of the factors), so AB does not equal BA and AB + BA does not equal 2AB.
6) A5 is invertible => TRUE
A5 = A4 * A = A3 * A * A = A2 * A * A *A = A * A * A * A * A
The product of invertible matrices is always invertible.
Greg K.
I meant #3****, sorry.12/15/21
Greg K.
#4 is not true, A+B is not always invertible. Take for example the 2x2 invertible matrices (1, 0)(0, 4) and (1,0)(0, -4) (format is (row1)(row2)). The addition of these matrices makes (2,0)(0,0), a matrix that is not invertible.12/15/21