Linear algebra, matrix 8 questions

[plshelp111]

Hello!
Im in desperate need of help, I need to upload 3 essays and worksheets for math, but ive stumbled across these questions which I cant solve. Please help, my whole year depends on it. Its translated from dutch.

AA.

 

[HallsofIvy]

Have you ever taken a course in "Linear Algebra"? Do you know what these words you are using mean?

Actually, I find the very first problem, 1a, peculiar! It asks whether it is true of false that, for A and B n by n matrices, \(\displaystyle (A\cdot B)= A^2\cdot B^2\). Obviously that is false! But I rather think it was intended to be the more interesting "\(\displaystyle (A\cdot B)^2= A^2\cdot B^2\)"- which is also false!

 

[plshelp111]

Have you ever taken a course in "Linear Algebra"? Do you know what these words you are using mean?

Actually, I find the very first problem, 1a, peculiar! It asks whether it is true of false that, for A and B n by n matrices, \(\displaystyle (A\cdot B)= A^2\cdot B^2\). Obviously that is false! But I rather think it was intended to be the more interesting "\(\displaystyle (A\cdot B)^2= A^2\cdot B^2\)"- which is also false!

It was an assignment for 11th grade 16-17 year old math, these questions were under linear algebra. Can you please help with the rest and explain why 1a is false?

 

[plshelp111]

Have you ever taken a course in "Linear Algebra"? Do you know what these words you are using mean?

Actually, I find the very first problem, 1a, peculiar! It asks whether it is true of false that, for A and B n by n matrices, \(\displaystyle (A\cdot B)= A^2\cdot B^2\). Obviously that is false! But I rather think it was intended to be the more interesting "\(\displaystyle (A\cdot B)^2= A^2\cdot B^2\)"- which is also false!

Wow yes it was supposed to be (A⋅B)²=A²⋅B²

 

[Steven G]

(AB)²= ABAB and

if this equals A²B².....................edited and corrected (offending party has been sent to the "corner" for 2 hrs.)​

we can multiply on the left side of both sides of the equation by A^-1and the the right hand side of both sides of the equation to by B^-1 to get AB=BA which is not always true. Why is that so hard?

(AB)^T_i,j= ???