Chapter 9

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

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 \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{rr}5 & 3 \\ -6 & -4\end{array}\right]$$

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

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. $$(2,0,4),\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$$

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 \(\mathrm{Av}_{2}\) $$A=\left[\begin{array}{ll}-2 & -1 \\ -2 & -1\end{array}\right]$$

Use the determinant to determine whether the matrix $$A=\left[\begin{array}{rr} -1 & 2 \\ 2 & -4 \end{array}\right]$$ 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. $$(-1,3,-2),\left[\begin{array}{r}-1 \\ -2 \\ 4\end{array}\right]$$

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

Suppose that $$ A=\left[\begin{array}{ll} 2 & 4 \\ 3 & 6 \end{array}\right] $$ (a) Compute det \(A\). Is \(A\) invertible? (b) Suppose that $$ X=\left[\begin{array}{l} x \\ y \end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right] $$ Write \(A X=B\) as a system of linear equations. (c) Show that if $$ B=\left[\begin{array}{l} 2 \\ 3 \end{array}\right] $$ then $$ A X=B $$ has infinitely many solutions. Graph the two straight lines associated with the corresponding system of linear equations, and explain why the system has infinitely many solutions. (d) Find a column vector $$ B=\left[\begin{array}{l} b_{1} \\ b_{2} \end{array}\right] $$ so that $$A X=B$$ has no solutions.

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

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{rr}-7 & 0 \\ 0 & 6\end{array}\right]$$

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 given points. \((5,4,-1)\) and \((2,0,3)\)

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{lr}1 & -3 \\ 0 & 2\end{array}\right]$$

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} 2 & 1 \\ -3 & -1 \end{array}\right] $$

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

Find the eigenvalues \(\lambda_{1}\) and \(\lambda_{2}\) for each matrix \(A\). $$A=\left[\begin{array}{rr}-1 & 4 \\ 0 & -2\end{array}\right]$$

Write down the inverse of \(A\). $$ A=\left[\begin{array}{ll} 1 & 2 \\ 1 & 3 \end{array}\right] $$

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 $$ A=\left[\begin{array}{ll} a & 0 \\ c & b \end{array}\right] $$

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} -1 & 4 \\ 5 & 0 \end{array}\right] $$

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 $$ A=\left[\begin{array}{ll} a & c \\ 0 & b \end{array}\right] $$

Write down the inverse of \(A\). $$ A=\left[\begin{array}{rr} -2 & -1 \\ 3 & 2 \end{array}\right] $$

Where do a plane through \((1,-1,2)\) and perpendicular to \(\left[\begin{array}{l}1 \\ 2 \\ 1\end{array}\right]\) and a line through the points \((0,-3,2)\) and \((1,-2,3)\) intersect?

(a) Show that the eigenvalues of the matrix \(A=\left[\begin{array}{ll}a & 0 \\\ 0 & c\end{array}\right]\) are \(\lambda_{1}=a\), and \(\lambda_{2}=c\). (b) Show that the corresponding eigenvectors are \(\mathbf{v}_{1}=\left[\begin{array}{l}1 \\ 0\end{array}\right]\) and \(\mathbf{v}_{2}=\left[\begin{array}{l}0 \\ 1\end{array}\right]\).

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 \(A X=\mathbf{0}\).

Where do a plane through \((2,0,-1)\) and perpendicular to \(\left[\begin{array}{r}-1 \\ 1 \\ 3\end{array}\right]\) and a line through the points \((1,0,-2)\) and \((1,-1,1)\) intersect?

Let $$ A=\left[\begin{array}{rr} -2 & -3 \\ -1 & 1 \end{array}\right] $$ Without explicitly computing the eigenvalues of \(A\), decide whether or not the real parts of both eigenvalues are negative.

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

Given a plane through \((0,-2,1)\) and perpendicular to \(\left[\begin{array}{r}-1 \\ 1 \\ -1\end{array}\right]\), find a line through \((5,-1,0)\) that is parallel to the plane.

Let $$ A=\left[\begin{array}{rr} 1 & 4 \\ -4 & -3 \end{array}\right] $$ Without explicitly computing the eigenvalues of \(A\), decide whether or not the real parts of both eigenvalues are negative.

Use the determinant to determine whether $$C=\left[\begin{array}{ll} 1 & 3 \\ 1 & 3 \end{array}\right]$$ is invertible. If it is invertible, compute its inverse. In either case, solve \(C X=\mathbf{0}\).

Let $$ A=\left[\begin{array}{ll} 0 & -1 \\ 2 & -1 \end{array}\right] $$ Without explicitly computing the eigenvalues of \(A\), decide whether or not the real parts of both eigenvalues are negative.

Use the determinant to determine whether $$D=\left[\begin{array}{ll} -3 & 6 \\ -4 & 8 \end{array}\right]$$ is invertible. If it is invertible, compute its inverse. In either case, solve \(D X=\mathbf{0}\).

Let $$ A=\left[\begin{array}{rr} 2 & 2 \\ 2 & -3 \end{array}\right] $$ Without explicitly computing the eigenvalues of \(A\), decide whether or not the real parts of both eigenvalues are negative.

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} 2 & -1 & -1 \\ 2 & 1 & 1 \\ -1 & 1 & -1 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} -2 & 5 \\ 2 & -3 \end{array}\right] $$ Without explicitly computing the eigenvalues of \(A\), decide whether or not the real parts of both eigenvalues are negative.

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} 1 & 2 & 1 \\ 0 & 1 & 2 \\ -1 & -1 & -1 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} -1 & 1 \\ 0 & 2 \end{array}\right] $$ (a) Show that $$ \mathbf{u}_{1}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \quad \text { and } \quad \mathbf{u}_{2}=\left[\begin{array}{l} 1 \\ 3 \end{array}\right] $$ are eigenvectors of \(A\) and that \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are linearly independent. (b) Represent $$ \mathbf{x}=\left[\begin{array}{r} 1 \\ -3 \end{array}\right] $$ as a linear combination of \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\). (c) Use your results in (a) and (b) to compute \(A^{20} \mathbf{x}\).

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & -1 \\ 0 & -2 & 0 \\ -1 & 1 & 2 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} -1 & -2 \\ -4 & 1 \end{array}\right] $$ (a) Show that $$ \mathbf{u}_{1}=\left[\begin{array}{r} 1 \\ -2 \end{array}\right] \quad \text { and } \quad \mathbf{u}_{2}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] $$ are eigenvectors of \(A\) and that \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\) are linearly independent. (b) Represent $$ \mathbf{x}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] $$ as a linear combination of \(\mathbf{u}_{1}\) and \(\mathbf{u}_{2}\). (c) Use your results in (a) and (b) to compute \(A^{10} \mathbf{x}\).

Find the inverse matrix to each given matrix if the inverse matrix exists. $$ A=\left[\begin{array}{rrr} -1 & 0 & 2 \\ -1 & -2 & 3 \\ 0 & 2 & -1 \end{array}\right] $$

Let $$ A=\left[\begin{array}{rr} -1 & 0 \\ 2 & 1 \end{array}\right] $$ Find $$ A^{15}\left[\begin{array}{l} 2 \\ 0 \end{array}\right] $$ without using a calculator.

Let $$ A=\left[\begin{array}{rr} 4 & 3 \\ 2 & -1 \end{array}\right] . $$ Find $$ A^{30}\left[\begin{array}{l} -4 \\ -2 \end{array}\right] $$ without using a calculator.

Let $$ A=\left[\begin{array}{rr} 5 & 7 \\ -2 & -4 \end{array}\right] $$ Find $$ A^{20}\left[\begin{array}{l} -3 \\ -2 \end{array}\right] $$ without using a calculator.