Square Matrices?

Question

Hi, I'm stuck on a question about square matrices for quite a long time. Any help would be greatly appreciated!

1. Let A and B be square matrices with the same size.
a) Give an example in which (A+B)^2 =/ (does not equal to) A^2 + 2AB + B^2
b) Fill in the blank to create a matrix identity that is valid for all choices of A and B. (A+B)^2 = A^2 + B^2 + _______.

2. Let A and B be square matrices with the same size.
a) Give an example in which (A+B)(A-B) =/ (does not equal to) A^2 - B^2
b) Fill in the blank to create a matrix identity that is valid for all choices of A and B. (A+B)(A-B) = ____________.

Thanks!

I can't seem to find any examples =\

Answer 1

The whole point of this exercise is to emphasize that matrix multiplication is NOT cummuntative.

Try
A =
1 0
1 0

and
B=
0 1
0 1

I think that you'll find that these two matices do not commute.

Answer 2

If you are only allowing for sub-matrices that consist of terms that are already adjacent, then for an n x n matrix, there would be n², 1x1's, (n-1)² 2x2's, (n-2)², 3x3's, ..., 1 nxn. This is a total of 1 + 2² + 3² + ... + n² = n(n+1)(2n+1)/6 such matrices. When I say that you are only allowing for matrices with terms that are already adjacent, I mean that you are not crossing out middle rows and columns. For example, with the 3x3 you've posted, you are not counting the 2x2 matrix [1 3] [7 9] obtained by deleting row 2 and column 2.

Answer 3

1.b) BA + AB

2.b) A^2 + BA - AB - B^2.

Examples for the a) parts should not be hard to find.