Answer:
Step-by-step explanation:add 4 to the side that has "13" then you will get "17" then divided -7 from x and "17"and your answer is x=idk
I think it’s
I’m not sure tho
(a) is false. Consider the identity matrix,
. Then
![(I+I^{-1})^3=(2I)^3=8I](https://tex.z-dn.net/?f=%28I%2BI%5E%7B-1%7D%29%5E3%3D%282I%29%5E3%3D8I)
bu
![I^3+I^{-1}=I+I=2I](https://tex.z-dn.net/?f=I%5E3%2BI%5E%7B-1%7D%3DI%2BI%3D2I)
(b) is false. Since
is invertible, we have
![ABA^{-1}=B\implies (AB)(A^{-1}A)=BA\implies AB=BA](https://tex.z-dn.net/?f=ABA%5E%7B-1%7D%3DB%5Cimplies%20%28AB%29%28A%5E%7B-1%7DA%29%3DBA%5Cimplies%20AB%3DBA)
but in general, matrix multiplication is non-commutative.
(c) is true. Let
; then
![B(A^{-1})^8=A^7(AA^{-1})(A^{-1})^7=A^7(A^{-1})^7](https://tex.z-dn.net/?f=B%28A%5E%7B-1%7D%29%5E8%3DA%5E7%28AA%5E%7B-1%7D%29%28A%5E%7B-1%7D%29%5E7%3DA%5E7%28A%5E%7B-1%7D%29%5E7)
![B(A^{-1})^8=A^6(AA^{-1})(A^{-1})^6=A^6(A^{-1})^6](https://tex.z-dn.net/?f=B%28A%5E%7B-1%7D%29%5E8%3DA%5E6%28AA%5E%7B-1%7D%29%28A%5E%7B-1%7D%29%5E6%3DA%5E6%28A%5E%7B-1%7D%29%5E6)
and so on, down to
![B(A^{-1})^8=AA^{-1}=I](https://tex.z-dn.net/?f=B%28A%5E%7B-1%7D%29%5E8%3DAA%5E%7B-1%7D%3DI)
which is to say
.
(d) is false. Consider
, so that
![(-I)+I=0](https://tex.z-dn.net/?f=%28-I%29%2BI%3D0)
which is singular (i.e. determinant is zero), and thus not invertible.
(e) is true. It follows from the distributivity of matrix multiplication:
![(I+A)(I+A^{-1})=I^2+AI+IA^{-1}+AA^{-1}=I+A+A^{-1}+I=2I+A+A^{-1}](https://tex.z-dn.net/?f=%28I%2BA%29%28I%2BA%5E%7B-1%7D%29%3DI%5E2%2BAI%2BIA%5E%7B-1%7D%2BAA%5E%7B-1%7D%3DI%2BA%2BA%5E%7B-1%7D%2BI%3D2I%2BA%2BA%5E%7B-1%7D)
(f) is false for the same reason as (b). Expanding the product gives
![(A+B)(A-B)=A^2+BA-AB-B^2](https://tex.z-dn.net/?f=%28A%2BB%29%28A-B%29%3DA%5E2%2BBA-AB-B%5E2)
but in general,
.