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The question is to find the values of \$v_1\$ and \$v_2\$:

enter image description here

I got the following matrix:

\begin{bmatrix} 0.64 < 38.66^{\circ} & 0.4 < -90^{\circ}\\ 0.85 < 28.07^{\circ}& 0.29<210.96^{\circ}\end{bmatrix} \begin{bmatrix} v_1 \\ v_2\end{bmatrix} $$=$$ \begin{bmatrix} 30<-90^{\circ} \\ 0\end{bmatrix}

Which gave me the answer: $$v_1=33.46 \cos(2t-211.49^{\circ})$$

Which is off from what the answer gives by .5 in the coefficient. Where did I go wrong?

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I get the following equations for the phasors:

$$I = v_1[1/R_1+j\omega C] - v_2 j\omega C\\ 0 = -v_1j\omega C + v_2[1/R_2+j(\omega C - 1/\omega L)]$$

The first equation is the same as you have in your matrix, but the second one is different. It just comes from adding up the currents at the point where \$v_2\$ is defined. The first coefficient of the second equation above is different from yours, but for the second coefficient I get the same magnitude (0.29155) but a different phase angle. By the way, even though I'm convinced that the equations above are correct, the final result is not equal to the one that's shown in your notes ...

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