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Other provided information: 'Let A and B be square matrices of the same size.'

The last question for my Linear Algebra assignment involves providing matrices $A$ and $B$ that satisfy the above requirement, as well as 'Give a valid identity for $(A + B)^2$' and I do not see any theorem or example from my text that helps me break this problem down. So far I have been guessing random matrices to no avail and I know there must be a methodical solution to this.

What do I need to know in order to approach this problem?

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    Guessing random size $2$ matrices should work. - good strategy is to make matrices out of $1$s and $0$s. – 2017-02-01
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    You need to include dollar signs (\$) to make MathJax do its work. – 2017-02-01
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    Essentially anything where multiplication is not commutative,e.g.. matrices. – 2017-02-01
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    See here:http://math.stackexchange.com/questions/699505/for-all-square-matrices-a-and-b-of-the-same-size-it-is-true-that-ab2?rq=1 – 2017-02-01

2 Answers 2

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You also ask what a valid identity for $(A+B)^2$ is. Because you asked:

What do I need to know in order to approach this problem?

I will first outline a few basic facts about matrix multiplication, which will help you.

  • In general, matrix multiplication is not commutative. That means that $AB \neq BA$. They can be equal, but they are not equal in general.
  • Matrix multiplication is distributive, so $(A+B)C=AC+BC$.

If the last applied three times on $(A+B)^2$, you will get the correct identity.

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take A = $\pmatrix{0 & -1\\-1 & 0}$ and B = $\pmatrix{1 & 0\\0 & 0}$. Basic idea is to provide A and B such that AB $\neq$BA

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    `\pmatrix{ a & b \\ c & d }` gives $\pmatrix{a & b \\ c & d}$ – 2017-02-01
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    You need to include dollar signs ($) to make MathJax (the formulas) do its work – 2017-02-01
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    yes!...Thanks for the help – 2017-02-01