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Prove that if the trace of a $2\times 2$ matrix over $\mathbb{R}$ is $0$, then it is a linear combination of matrices of the form $XY-YX$, where $X$ and $Y$ denote arbitrary $2\times 2$ matrices over $\mathbb{R}$.

I understand that for the trace to be $0$ for a matrix

$$ \left[ \begin{array}{ c c } a & b \\ c & d \end{array} \right] $$

either $a=-d$ or $a=d=0$. Trying to work backwards by finding an arbitrary $XY-YX$ for any matrix by brute force hasn't proved useful for me so far. Perhaps there's a better solution?

Any help greatly appreciated!

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    Notice that your second case ($a = d = 0$) is subsumed by the first. Hint: is the set of traceless matrices a subspace? – 2017-01-17

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Hint: Try computing $XY-YX$ when $X$ and $Y$ are very simple matrices; say matrices with only one nonzero component. What matrices can you produce this way? What matrices do you get when you take linear combinations of these matrices?