Chapter 9

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 \(A I_{2}=I_{2} A=A\).

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

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 \(A I_{3}=I_{3} A=A\).

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). $$ \begin{array}{r} 2 x_{1}+3 x_{2}-x_{3}=0 \\ 3 x_{2}+x_{3}=1 \\ x_{1}-x_{3}=2 \end{array} $$

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

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

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

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

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

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 \((1,-1,2)\) and perpendicular to \([-1,1,2]^{\prime}\).

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

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

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. $$(3,-4),\left[\begin{array}{l}1 \\ 1\end{array}\right]$$

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}-2 \\\ -3\end{array}\right]\) counterclockwise 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. $$(-1,-2),\left[\begin{array}{r}1 \\ -2\end{array}\right]$$

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

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Find the inverse (if it exists) of \(A\).

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. $$(-1,4),\left[\begin{array}{l}2 \\ 3\end{array}\right]$$

Use a rotation matrix to rotate the vector \(\left[\begin{array}{l}1 \\\ 2\end{array}\right]\) counterclockwise by the angle \(\pi / 6\).

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Find the inverse (if it exists) of \(B\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. \((-1,2)\) and \((3,0)\)

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}5 \\\ -3\end{array}\right]\) clockwise by the angle \(\pi / 2\).

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Show that \(\left(A^{-1}\right)^{-1}=A\)

Find the parametric equation of the line in the \(x-y\) plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. \((2,1)\) and \((3,5)\)

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

Let $$\boldsymbol{A}=\left[\begin{array}{rr} -\mathbf{1} & \mathbf{1} \\ \mathbf{2} & \mathbf{3} \end{array}\right], \quad \boldsymbol{B}=\left[\begin{array}{ll} \mathbf{2} & \mathbf{0} \\ \mathbf{3} & \mathbf{2} \end{array}\right]$$ Show that \((A B)^{-1}=B^{-1} A^{-1}\).

Find the parametric equation of the line in the \(x-y\) plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. \((1,-3)\) and \((4,-3)\)

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}2 & 3 \\ 0 & -1\end{array}\right]$$

Find the inverse (if it exists) of $$C=\left[\begin{array}{ll} 1 & 2 \\ 3 & 6 \end{array}\right]$$

Find the parametric equation of the line in the \(x-y\) plane that goes through the given points. Then eliminate the parameter to find the equation of the line in standard form. \((2,3)\) and \((-1,-4)\)

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}0 & 0 \\ 1 & -3\end{array}\right]$$

Find the inverse (if it exists) of $$I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]$$

Parameterize the equation of the line given in standard form. $$3 x+2 y-1=0$$

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}1 & 0 \\ 0 & -1\end{array}\right]$$

Suppose that $$A=\left[\begin{array}{rr} -1 & 0 \\ 2 & -1 \end{array}\right] \text { and } D=\left[\begin{array}{l} -2 \\ -5 \end{array}\right]$$ Find \(X\) such that \(A X=D\) by (a) solving the associated system of linear equations and (b) using the inverse of \(A\).

Parameterize the equation of the line given in standard form. $$x-2 y+5=0$$

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}-1 & 0 \\ 0 & 2\end{array}\right]$$

(a) Show that if \(X=A X+D\), then $$X=(I-A)^{-1} D$$ provided that \(I-A\) is invertible. (b) Suppose that $$A=\left[\begin{array}{rr} 3 & 2 \\ 0 & -1 \end{array}\right] \text { and } D=\left[\begin{array}{r} -2 \\ 2 \end{array}\right]$$ Compute \((I-A)^{-1}\), and use your result in (a) to compute \(X\).

Parameterize the equation of the line given in standard form. $$2 x+y-3=0$$

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}-1 & 2 \\ 4 & 1\end{array}\right]$$

Use the determinant to determine whether the matrix $$A=\left[\begin{array}{rr} 2 & -1 \\ -1 & 3 \end{array}\right] $$ is invertible.

Parameterize the equation of the line given in standard form. $$2 x-y+4=0$$

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\), and graph the lines together with the eigenvectors \(\mathrm{v}_{1}\) and \(\mathrm{v}_{2}\) and the vectors \(\mathrm{Av}_{1}\) and \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}-1 & 0 \\ 4 & 3\end{array}\right]$$