Chapter 8

In Problems, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right) $$

Use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{rrr} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{array}\right) $$

Determine whether the given matrix is orthogonal. $$ \left(\begin{array}{rrr} 1 & -1 & 1 \\ 1 & -1 & -1 \\ 1 & 2 & 0 \end{array}\right) $$

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{rrr} 1 & 2 & 3 \\ 0 & -4 & 2 \\ -1 & 5 & 1 \end{array}\right) $$

Solve the given system of equations by Cramer's rule. $$ \begin{array}{r} u+2 v+\quad w=8 \\ 2 u-2 v+2 w=7 \\ u-4 v+3 w=1 \end{array} $$

Determine whether the given matrices are equal. $$ \left(\begin{array}{cc} \sqrt{(-2)^{2}} & 1 \\ 2 & \frac{2}{8} \end{array}\right),\left(\begin{array}{rr} -2 & 1 \\ 2 & \frac{1}{4} \end{array}\right) $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(x_{1}-x_{2}-x_{3}=8\) \(x_{1}-x_{2}+x_{3}=3\)

In Problems 1-20, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right) $$

In Problems 7-10, use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{rrr} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{array}\right) $$

$$ \text { In Problems } 5-10 \text {, determine whether the given matrix is orthogonal. } $$ $$ \left(\begin{array}{rrr} 1 & -1 & 1 \\ 1 & -1 & -1 \\ 1 & 2 & 0 \end{array}\right) $$

In Problems 7-22, find the eigenvalues and eigenvectors of the given matrix. Using Theorem 8.8.2 or (6), state whether the matrix is singular or nonsingular. $$ \left(\begin{array}{rr} -8 & -1 \\ 16 & 0 \end{array}\right) $$

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{array}{r} u+2 v+w=8 \\ 2 u-2 v+2 w=7 \\ u-4 v+3 w=1 \end{array} $$

In Problems 9-14, evaluate the determinant of the given matrix. $$ (-7) $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} x_{1}-x_{2}-x_{3}=8 \\ x_{1}-x_{2}+x_{3}=3 \\ -x_{1}+x_{2}+x_{3}=4 \end{array} $$

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{cc} \sqrt{(-2)^{2}} & 1 \\ 2 & \frac{2}{8} \end{array}\right),\left(\begin{array}{rr} -2 & 1 \\ 2 & \frac{1}{4} \end{array}\right) $$

In Problems, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rr} 1 & 2 \\ -\frac{1}{2} & 1 \end{array}\right) $$

Use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{rrr} 0 & 0 & -4 \\ 0 & -4 & 0 \\ -4 & 0 & 15 \end{array}\right) $$

Determine whether the given matrix is orthogonal. $$ \left(\begin{array}{rrrr} 0 & \frac{8}{17} & 0 & -\frac{15}{17} \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & \frac{15}{17} & 0 & \frac{8}{17} \end{array}\right) $$

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{rrr} 2 & -1 & 5 \\ 3 & 0 & -2 \\ 1 & 4 & 0 \end{array}\right) $$

Solve the given system of equations by Cramer's rule. $$ \begin{aligned} 4 x+3 y+2 z &=8 \\ -x+& 2 z=12 \\ 3 x+2 y+z &=3 \end{aligned} $$

State the appropriate theorem(s) in this section that justifies the given equality. Do not expand the determinants by cofactors. $$ \left|\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right|=\left|\begin{array}{llll} 0 & 0 & 0 & 1 \\ 0 & 0 & 2 & 0 \\ 0 & 3 & 0 & 0 \\ 4 & 0 & 0 & 0 \end{array}\right| $$

Determine whether the given matrices are equal. $$ \left(\begin{array}{rr} \frac{1}{8} & \frac{1}{5} \\ \sqrt{2} & 1 \end{array}\right),\left(\begin{array}{lr} 0.125 & 0.2 \\ 1.414 & 1 \end{array}\right) $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(3 x_{1}+x_{2}=4\) \(4 x_{1}+3 x_{2}=-3\) \(2 x_{1}-x_{2}=11\)

In Problems 1-20, fill in the blanks or answer true/false. Suppose \(\mathbf{A}\) is a \(3 \times 3\) matrix such that \(\operatorname{det} \mathbf{A}=2 .\) If \(\mathbf{B}=10 \mathbf{A}\) and \(\mathbf{C}=-\mathbf{B}^{-1}\), then \(\operatorname{det} \mathbf{C}=\) _________.

In Problems 1-20, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} 1 & 2 \\ -\frac{1}{2} & 1 \end{array}\right) $$

In Problems 7-10, use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{rrr} 0 & 0 & -4 \\ 0 & -4 & 0 \\ -4 & 0 & 15 \end{array}\right) $$

$$ \text { In Problems } 5-10 \text {, determine whether the given matrix is orthogonal. } $$ $$ \left(\begin{array}{rrrr} 0 & \frac{8}{17} & 0 & -\frac{15}{17} \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & \frac{15}{17} & 0 & \frac{8}{17} \end{array}\right) $$

In Problems 7-22, find the eigenvalues and eigenvectors of the given matrix. Using Theorem 8.8.2 or (6), state whether the matrix is singular or nonsingular. $$ \left(\begin{array}{ll} 1 & 1 \\ \frac{1}{4} & 1 \end{array}\right) $$

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} 4 x+3 y+2 z &=8 \\ -x+\quad 2 z &=12 \\ 3 x+2 y+z &=3 \end{aligned} $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} &3 x_{1}+x_{2}=4 \\ &4 x_{1}+3 x_{2}=-3 \\ &2 x_{1}-x_{2}=11 \end{aligned} $$

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{rr} \frac{1}{8} & \frac{1}{5} \\ \sqrt{2} & 1 \end{array}\right),\left(\begin{array}{rr} 0.125 & 0.2 \\ 1.414 & 1 \end{array}\right) $$

In Problems, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rrr} 1 & 0 & 1 \\ 0 & -1 & 3 \\ 0 & 0 & 2 \end{array}\right) $$

Use the inverse power method to find the eigenvalue of least magnitude for the given matrix. $$ \left(\begin{array}{ll} 1 & 1 \\ 3 & 4 \end{array}\right) $$

Show that the given matrix has an eigenvalue \(\lambda_{1}\) of multiplicity two. As a consequence, the equations \(\lambda^{m}=c_{0}+c_{1} \lambda(\) Problem 11\()\) and \(\lambda^{m}=c_{0}+c_{1} \lambda+c_{2} \lambda^{2}\) (Problem 12) do not yield enough independent equations to form a system for determining the coefficients \(c_{i}\). Use the derivative (with respect to \(\lambda\) ) of each of these equations evaluated at \(\lambda_{1}\) as the extra needed equation to form a system. Compute \(\mathbf{A}^{m}\) and use this result to compute the indicated power of the matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rr} 7 & 3 \\ -3 & 1 \end{array}\right) ; \quad m=6 $$

In Problems, find the eigenvalues and eigenvectors of the given matrix. Using Theorem \(8.8 .2\) or (6), state whether the matrix is singular or nonsingular. $$ \left(\begin{array}{ll} -1 & 2 \\ -5 & 1 \end{array}\right) $$

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{rrr} 3 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -2 \end{array}\right) $$

Use Cramer's rule to determine the solution of the system $$ \begin{array}{r} (2-k) x_{1}+\quad k x_{2}=4 \\ k x_{1}+(3-k) x_{2}=3 \end{array} $$

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{rr} 3 & 5 \\ -1 & 4 \end{array}\right) $$

Evaluate the determinant of the given matrix using the result \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\\ c_{1} & c_{2} & c_{3}\end{array}\right|=5\). $$ \mathbf{A}=\left(\begin{array}{lll} a_{3} & a_{2} & a_{1} \\ b_{3} & b_{2} & b_{1} \\ c_{3} & c_{2} & c_{1} \end{array}\right) $$

Determine the values of \(x\) and \(y\) for which the matrices are equal. $$ \left(\begin{array}{rr} 1 & x \\ y & -3 \end{array}\right),\left(\begin{array}{cc} 1 & y-2 \\ 3 x-2 & -3 \end{array}\right) $$

Determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 1,2,3\rangle, \mathbf{u}_{2}=\langle 1,0,1\rangle, \mathbf{u}_{3}=\langle 1,-1,5\rangle $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(\begin{aligned} 2 x_{1}+2 x_{2} &=0 \\\\-2 x_{1}+x_{2}+x_{3} &=0 \\ 3 x_{1}+x_{3} &=0 \end{aligned}\)

In Problems 1-20, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrr} 1 & 0 & 1 \\ 0 & -1 & 3 \\ 0 & 0 & 2 \end{array}\right) $$

In Problems 7-22, find the eigenvalues and eigenvectors of the given matrix. Using Theorem 8.8.2 or (6), state whether the matrix is singular or nonsingular. $$ \left(\begin{array}{ll} -1 & 2 \\ -5 & 1 \end{array}\right) $$

In Problems 11-14, evaluate the determinant of the given matrix using the result $$ \left|\begin{array}{lll} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{array}\right|=5 $$ $$ \mathbf{A}=\left(\begin{array}{lll} a_{3} & a_{2} & a_{1} \\ b_{3} & b_{2} & b_{1} \\ c_{3} & c_{2} & c_{1} \end{array}\right) $$

In Problems 9-14, evaluate the determinant of the given matrix. $$ \left(\begin{array}{rr} 3 & 5 \\ -1 & 4 \end{array}\right) $$

In Problems \(11-14\), determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 1,2,3\rangle, \mathbf{u}_{2}=\langle 1,0,1\rangle, \mathbf{u}_{3}=\langle 1,-1,5\rangle $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} 2 x_{1}+2 x_{2}=0 \\ -2 x_{1}+x_{2}+x_{3}=0 \\ 3 x_{1}+\quad x_{3}=0 \end{array} $$

In Problems 11 and 12, determine the values of \(x\) and \(y\) for which the matrices are equal. $$ \left(\begin{array}{rr} 1 & x \\ y & -3 \end{array}\right),\left(\begin{array}{cc} 1 & y-2 \\ 3 x-2 & -3 \end{array}\right) $$

In Problems, determine whether the given matrix \(\mathbf{A}\) is diagonalizable. If so, find the matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rrr} 1 & 2 & 2 \\ 2 & 3 & -2 \\ -5 & 3 & 8 \end{array}\right) $$