3x3 matrices examples to meet properties

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Solution 1

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 $B = A$.

Solution 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.

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Jill A Ray
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Jill A Ray

Going to school for Math Education.

Updated on August 01, 2022

Comments

  • Jill A Ray
    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
    Ben Grossmann over 9 years
    note that OP has requested $3 \times 3$ examples
  • Pedro
    Pedro over 9 years
    @Omnomnomnom Easily fixed by extending with a diagonal $1$.
  • qwr
    qwr over 9 years
    Lazy = clean and simple!
  • Rebecca J. Stones
    Rebecca J. Stones over 9 years
    Oops. I'll fix that. Thanks.