Give example of two non-zero matrices A and B such that A^2=0, B^2=0 but
AxB and BxA do not equal zero.
To find two non-zero matrices A and B such that A^2=0, B^2=0, but AxB and BxA are non-zero, we can construct the matrices as follows:
Let A be a 2x2 matrix with all elements equal to 0, except the (1,2) element which is 1.
A = | 0 1 |
| 0 0 |
Then, when we compute A^2, we get:
A^2 = A x A = | 0 1 | x | 0 1 | = | 0 0 |
| 0 0 | | 0 0 |
As you can see, A^2 is a matrix of all zeros.
Now, let B be a 2x2 matrix with all elements equal to 0, except the (2,1) element which is 1.
B = | 0 0 |
| 1 0 |
When we compute B^2, we get:
B^2 = B x B = | 0 0 | x | 0 0 | = | 0 0 |
| 1 0 | | 0 0 |
Again, B^2 is a matrix of all zeros.
However, when we compute AxB and BxA, we get:
AxB = | 0 1 | x | 0 0 | = | 0 0 |
| 0 0 | | 0 0 |
BxA = | 0 0 | x | 0 1 | = | 0 0 |
| 1 0 | | 0 0 |
As you can see, both AxB and BxA are non-zero matrices.