91Ó°ÊÓ

Write each system in matrix form. (There is no need to solve the systems). $$ \begin{array}{r} 2 x_{1}-x_{2}=4 \\ -x_{1}+2 x_{2}=3 \\ 3 x_{1}=4 \end{array} $$

Find the equation of the plane through \((0,0,0)\) and perpendicular to \([1,0,0]\) '.

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}4 \\\ -1\end{array}\right]\) counterclockwise by the angle \(\pi / 6\).

Write each system in matrix form. (There is no need to solve the systems). $$ \begin{array}{r} x_{1}-3 x_{2}+x_{3}=1 \\ -2 x_{1}+x_{2}-x_{3}=0 \end{array} $$

Find the equation of the plane through \((1,-1,2)\) and perpendicular to \([-1,1,2]^{\prime}\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$(1,-1),\left[\begin{array}{l}2 \\ 1\end{array}\right]$$

Show that the inverse of $$A=\left[\begin{array}{ll} 2 & 1 \\ 1 & 3 \end{array}\right]$$ is $$B=\frac{1}{5}\left[\begin{array}{rr} 3 & -1 \\ -1 & 2 \end{array}\right]$$

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}5 \\\ 2\end{array}\right]\) clockwise by the angle \(45^{\circ}\).

Show that the inverse of $$A=\left[\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array}\right]$$ is $$A^{-1}=\left[\begin{array}{ccc} \frac{3}{4} & \frac{1}{2} & \frac{1}{4} \\ \frac{1}{2} & 1 & \frac{1}{2} \\ \frac{1}{4} & \frac{1}{2} & \frac{3}{4} \end{array}\right]$$

Find the parametric equation of the line in the \(x-y\) plane that goes through the indicated point in the direction of the indicated vector. $$(3,-4),\left[\begin{array}{l}1 \\ 1\end{array}\right]$$