91Ó°ÊÓ

Let $$A=\left[\begin{array}{rr} 1 & 1 \\ 1 & -2 \end{array}\right] \quad \text { and } \quad I_{2}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]$$ Show that \(A I_{2}=I_{2} A=A\).

Let \(A=\left[\begin{array}{rrr}1 & 3 & 0 \\ 0 & 0 & -2 \\ -1 & 1 & 1\end{array}\right]\) and \(I_{3}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]\) Show that \(A I_{3}=I_{3} A=A\).

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\left[\begin{array}{rr}0 & 1 \\ -1 & 0\end{array}\right]$$

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

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

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\frac{1}{2}\left[\begin{array}{rr}\sqrt{3} & -1 \\ 1 & \sqrt{3}\end{array}\right]$$

Write each system in matrix form. (There is no need to solve the systems). $$ \begin{array}{r} 2 x_{2}-x_{1}=x_{3} \\ 4 x_{1}+x_{3}=7 x_{2} \\ x_{2}-x_{1}=x_{3} \end{array} $$

Give a geometric interpretation of the map \(\mathrm{x} \mapsto\) Ax for each given map \(\mathrm{A}\). $$A=\frac{1}{2}\left[\begin{array}{lr}\sqrt{2} & -\sqrt{2} \\ \sqrt{2} & \sqrt{2}\end{array}\right]$$

Find the equation of the plane through \((1,0,-3)\) and perpendicular to \([1,-2,-1]^{\prime}\).

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}-1 \\\ 2\end{array}\right]\) counterclockwise by the angle \(\pi / 3\).