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 such matrices, we need to consider the properties of matrix multiplication and the conditions given.
Let's start by considering the condition A^2 = 0. This means that the matrix A, when squared or multiplied by itself, results in the zero matrix.
Similarly, for B^2 = 0, the matrix B multiplied by itself should also yield the zero matrix.
Now, in order for AxB and BxA to not equal zero, we need to find matrices A and B that have non-zero entries but still satisfy the conditions A^2 = 0 and B^2 = 0.
We can achieve this by using matrices that have non-zero entries in certain positions, while the other entries are zero.
For example, let's consider the following matrices:
A = [[0, 1],
[0, 0]]
B = [[0, 0],
[1, 0]]
Now, let's verify the conditions:
A^2 = [[0, 1],
[0, 0]]
x [[0, 1],
[0, 0]]
= [[0, 0],
[0, 0]]
Since A^2 equals the zero matrix, the condition A^2 = 0 is satisfied.
Next, let's calculate AxB and BxA:
AxB = [[0, 1],
[0, 0]]
x [[0, 0],
[1, 0]]
= [[1, 0],
[0, 0]]
BxA = [[0, 0],
[1, 0]]
x [[0, 1],
[0, 0]]
= [[0, 0],
[0, 0]]
As you can see, AxB is not equal to the zero matrix, while BxA is equal to the zero matrix.
Therefore, the matrices A and B = [[0, 1], [0, 0]] and [[0, 0], [1, 0]], respectively, satisfy the conditions A^2 = 0, B^2 = 0, and AxB ≠ 0, BxA = 0.