Chapter 9

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

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

In Problems \(47-50\), 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. $$ (-1,2) \text { and }(3,4) $$

Use a rotation matrix to rotate the vector \(\left[\begin{array}{r}5 \\\ -3\end{array}\right]\) clockwise by the angle \(\pi / 7\).

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ 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. $$ (2,1) \text { and }(3,5) $$

$$ A=\left[\begin{array}{rr} -1 & 1 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & -2 \\ 3 & 2 \end{array}\right] $$ Find the inverse (if it exists) of $$ C=\left[\begin{array}{ll} 2 & 4 \\ 3 & 6 \end{array}\right] $$

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. \((1 .-3)\) and \((4,0)\)

In Problems 49-56, find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} 2 & 3 \\ 0 & -1 \end{array}\right] $$

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

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 \((-1,-4)\)

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

Suppose that $$ A=\left[\begin{array}{rr} -1 & 0 \\ 2 & -3 \end{array}\right] \text { and } D=\left[\begin{array}{l} -2 \\ -5 \end{array}\right] $$ Find \(X\) such that \(A X=D\) by (a) solving the associated system of linear equations and (b) using the inverse of \(A\).

In Problems \(51-54\), parameterize the equation of the line given in standard form. $$ 3 x+4 y-1=0 $$

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

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

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

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

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. $$ 2 x+y-3=0 $$

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

Use the determinant to determine whether the matrix $$ A=\left[\begin{array}{rr} -1 & 3 \\ 0 & 3 \end{array}\right] $$ is invertible.

Parameterize the equation of the line given in standard form. $$ x-5 y+7=0 $$

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} 3 & 6 \\ -1 & -4 \end{array}\right] $$

Use the determinant to determine whether the matrix $$ A=\left[\begin{array}{ll} 4 & -1 \\ 8 & -2 \end{array}\right] $$ is invertible.

In Problems \(55-58\), find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the indicated point in the direction of the indicated vector. $$ (1,-1,2),\left[\begin{array}{r} 1 \\ -2 \\ 1 \end{array}\right] $$

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left|\begin{array}{ll} 2 & 1 \\ 2 & 3 \end{array}\right| $$

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

Find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the indicated point in the direction of the indicated vector. $$ (2,0,4),\left[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right] $$

In Problems , find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) and corresponding eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) for each matrix A. Determine the equations of the lines through the origin in the direction of the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), and graph the lines together with the eigenvectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\) and the vectors \(A \mathbf{v}_{1}\) and \(A \mathbf{v}_{2}\). $$ A=\left[\begin{array}{rr} -3 & -0.5 \\ 7 & 1.5 \end{array}\right] $$

Suppose that $$ A=\left[\begin{array}{ll} 2 & 4 \\ 3 & 6 \end{array}\right] $$ (a) Compute det \(A\). Is \(A\) invertible?

Find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the indicated point in the direction of the indicated vector. $$ (-1,3,-2),\left[\begin{array}{r} -1 \\ -2 \\ 4 \end{array}\right] $$

$$ \begin{array}{l} \text { In Problems 57-60, find the eigenvalues } \lambda_{1} \text { and } \lambda_{2} \text { for each matrix }\\\ A \end{array} $$ $$ A=\left[\begin{array}{ll} 4 & 0 \\ 0 & 3 \end{array}\right] $$

Suppose that $$ A=\left[\begin{array}{ll} a & 8 \\ 2 & 4 \end{array}\right], \quad X=\left[\begin{array}{l} x \\ y \end{array}\right], \quad \text { and } \quad B=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right] $$ (a) Show that when \(a \neq 4, A X=B\) has exactly one solution. (b) Suppose \(a=4 .\) Find conditions on \(b_{1}\) and \(b_{2}\) such that \(A X=\) \(B\) has (i) infinitely many solutions and (ii) no solutions. (c) Explain your results in (a) and (b) graphically.

Find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the indicated point in the direction of the indicated vector. $$ (2,1,-3),\left[\begin{array}{r} 3 \\ -1 \\ 2 \end{array}\right] $$

$$ \begin{array}{l} \text { In Problems , find the eigenvalues } \lambda_{1} \text { and } \lambda_{2} \text { for each matrix }\\\ A \end{array} $$ $$ A=\left[\begin{array}{rr} -7 & 0 \\ 0 & 6 \end{array}\right] $$

Use the determinant to find the inverse of \(A\) $$ A=\left[\begin{array}{rr} 2 & 1 \\ 3 & -1 \end{array}\right] $$

In Problems \(59-62\), find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the given points. \((5,4,-1)\) and \((2,0,3)\)

$$ \begin{array}{l} \text { In Problems , find the eigenvalues } \lambda_{1} \text { and } \lambda_{2} \text { for each matrix }\\\ A \end{array} $$ $$ A=\left[\begin{array}{rr} 1 & -3 \\ 0 & 2 \end{array}\right] $$

Use the determinant to find the inverse of \(A\) $$ A=\left[\begin{array}{ll} 1 & 2 \\ 0 & 3 \end{array}\right] $$

Find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the given points. \((2,0,-3)\) and \((4,1,0)\)

$$ \begin{array}{l} \text { In Problems , find the eigenvalues } \lambda_{1} \text { and } \lambda_{2} \text { for each matrix }\\\ A \end{array} $$ $$ \text { 60. } A=\left[\begin{array}{rr} -1 & 4 \\ 0 & -2 \end{array}\right] $$

Use the determinant to find the inverse of \(A\) $$ A=\left[\begin{array}{rr} -1 & 4 \\ 5 & 1 \end{array}\right] $$

Find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the given points. \((2,-3,1)\) and \((-5,2,1)\)

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for $$ A=\left[\begin{array}{ll} a & 0 \\ c & b \end{array}\right] $$

Use the determinant to find the inverse of \(A\) $$ A=\left[\begin{array}{rr} -2 & 1 \\ 3 & 2 \end{array}\right] $$

Find the parametric equation of the line in \(x-\) \(y-z\) space that goes through the given points. \((1,0,4)\) and \((3,2,0)\)

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for $$ A=\left[\begin{array}{ll} a & c \\ 0 & b \end{array}\right] $$

Use the determinant to determine whether $$ A=\left[\begin{array}{rr} 1 & -1 \\ 0 & 2 \end{array}\right] $$ is invertible. If it is invertible, compute its inverse. In either case, solve \(A X=\mathbf{0}\).

Use the determinant to determine whether $$ B=\left[\begin{array}{ll} 1 & 1 \\ 2 & 1 \end{array}\right] $$ is invertible. If it is invertible, compute its inverse. In either case, solve \(B X=\mathbf{0}\).