# How to Simplify this to the given answer? (matrix equation, trig functions)

Can someone help me to simplify $$\Bigg( \begin{matrix} 5\cos t &5\sin t \\2\cos t+\sin t & 2\sin t-\cos t\end{matrix} \Bigg) \Bigg( \begin{matrix} u_1 \\u_2\end{matrix} \Bigg) =\Bigg( \begin{matrix} -\cos t \\ \phantom{-} \sin t \end{matrix} \Bigg)$$ to

$${u_1}={1\over5} \big(2-3\cos2t+\sin2t)$$ $$u_2={1\over5} \big(-1-\cos2t-3\sin2t)?$$

Thank you.

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What have you tried? –  apnorton Jan 7 '13 at 13:22
this is a part of a problem, –  Miranda Miri Jan 7 '13 at 13:25

$$AX=B$$ $$X=A^{-1}B$$
Where $X$ is the variable/unknown vector, $A$ is the coefficient matrix, and $B$ is the solution vector.
You could try the effect of multiplying the first row by $\cos t$ and the second row by $2 \cos t - \sin t$ - but whichever way you do it, it is essentially the same problem, and the determinant of the matrix you have comes out as an easy number - so the inverse can be written down. Why use more steps than you need to solve the problem? –  Mark Bennet Jan 7 '13 at 14:09