3x3 matrices examples to meet properties
Solution 1
Because being lazy is an artform, the simplest examples I can think of:
a)$A=\pmatrix{0&1&0\\0&0&0\\0&0&0}$. $B = \pmatrix{0&0&0\\1&0&0\\0&0&0}$
b) $A$ as above and $B = 2A$
c) $A$ as above. $B = 2A$ and $C = 3A$.
d) $A$ as above and $B = A$.
Solution 2
Hints:
Just generate two random matrices and check whether $AB=BA$ is true (if it is, start again). It unlikely to happen by chance.
Take one of them as $2$ times the identity matrix.
Try $$A=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$ and $$B=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \quad \text{and} \quad C=\begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}$$
Try $BC$ above.
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Comments

Jill A Ray over 1 year
I need to find 3x3 matrices that meet the following:
a) AB not equal to BA b) AB=BA but A not equal to B c)AB=AC but B does not equal C d) AB is the zero matrix but A nor B is the zero matrix
None can be the identity or zero matrix.
I've been playing with the hint etc but still struggling. I must not be doing something right, help! I'm not getting something, not sure what.
Thanks for help.

Ben Grossmann over 9 yearsnote that OP has requested $3 \times 3$ examples

Pedro over 9 years@Omnomnomnom Easily fixed by extending with a diagonal $1$.

qwr over 9 yearsLazy = clean and simple!

Rebecca J. Stones over 9 yearsOops. I'll fix that. Thanks.