3x3 matrices examples to meet properties

Question

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.

Answer 1

a)

$A=\left( \begin{array}{ccc} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 5 & 6 & 7 \\ \end{array} \right)$

$B=\left( \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ \end{array} \right)$

$A\cdot B =\left( \begin{array}{ccc} 30 & 36 & 42 \\ 54 & 66 & 78 \\ 78 & 96 & 114 \\ \end{array} \right)$

$B\cdot A=\left( \begin{array}{ccc} 22 & 28 & 34 \\ 49 & 64 & 79 \\ 76 & 100 & 124 \\ \end{array} \right)$

Answer 2

Hints:

1. Just generate two random matrices and check whether $AB=BA$ is true (if it is, start again). It unlikely to happen by chance.

2. Take one of them as $2$ times the identity matrix.

3. Try $$A=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$ and $$B=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \quad \text{and} \quad C=\begin{bmatrix} 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}$$

4. Try $BC$ above.

Answer 3

Because being lazy is an art-form, 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 $B = A$.