Chapter 8

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

In Problems, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A} .\) Give the corresponding eigenvalue. $$ \mathbf{A}=\left(\begin{array}{rrr} -1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 3 & -1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} -1 \\ 4 \\ 3 \end{array}\right) $$

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

Solve the given system of equations by Cramer's rule. $$ \begin{gathered} 5 r+4 s=-1 \\ 10 r-6 s=5 \end{gathered} $$

Suppose $$ \mathbf{A}=\left(\begin{array}{rrrr} 0 & 2 & 4 & 0 \\ 1 & 2 & -2 & 3 \\ 5 & 1 & 0 & -1 \\ 1 & 1 & 1 & 2 \end{array}\right) . $$ Evaluate the indicated minor determinant or cofactor. $$ M_{41} $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(x_{1}+2 x_{2}-x_{3}=0\) \(2 x_{1}+x_{2}+2 x_{3}=9\) \(x_{1}-x_{2}+x_{3}=3\)

State the size of the given matrix. $$ \left(\begin{array}{r} 1 \\ 5 \\ -6 \\ 0 \\ 7 \\ -10 \\ 2 \\ 12 \end{array}\right) $$

In Problems \(1-6\), find the least squares line for the given data. $$ (1,2),(2,2.5),(3,1),(4,1.5),(5,2),(6,3.2),(7,5) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} -5 & -3 \\ 5 & 11 \end{array}\right) $$

In Problems 3-6, use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. $$ \left(\begin{array}{lll} 3 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \end{array}\right) $$

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

In Problems 1-6, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A}\). Give the corresponding eigenvalue. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rrr} -1 & 1 & 0 \\ 1 & 2 & 1 \\ 0 & 3 & -1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} -1 \\ 4 \\ 3 \end{array}\right) \\ &\mathbf{K}_{2}=\left(\begin{array}{l} 1 \\ 4 \\ 3 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{l} 3 \\ 1 \\ 4 \end{array}\right) \end{aligned} $$

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} 5 r+4 s &=-1 \\ 10 r-6 s &=5 \end{aligned} $$

In Problems 5-8, suppose $$ \mathbf{A}=\left(\begin{array}{rrrr} 0 & 2 & 4 & 0 \\ 1 & 2 & -2 & 3 \\ 5 & 1 & 0 & -1 \\ 1 & 1 & 1 & 2 \end{array}\right) $$ \text { Evaluate the indicated minor determinant or cofactor. } $$ M_{41} $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} x_{1}+2 x_{2}-x_{3}=0 \\ 2 x_{1}+x_{2}+2 x_{3}=9 \\ x_{1}-x_{2}+x_{3}=3 \end{array} $$

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{r} 1 \\ 5 \\ -6 \\ 0 \\ 7 \\ -10 \\ 2 \\ 12 \end{array}\right) $$

In an experiment, the following correspondence was found between temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) and kinematic viscosity \(v\) (in Centistokes) of an oil with a certain additive: $$ \begin{array}{|lrrrrrr|} \hline \boldsymbol{T} & 20 & 40 & 60 & 80 & 100 & 120 \\ \hline \boldsymbol{v} & 220 & 200 & 180 & 170 & 150 & 135 \\ \hline \end{array} $$ Find the least squares line \(v=a T+b\). Use this line to estimate the viscosity of the oil at \(T=140\) and \(T=160\)

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{ll} \frac{1}{2} & \frac{1}{6} \\ \frac{1}{6} & \frac{1}{2} \end{array}\right) $$

Use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{ll} 3 & 2 \\ 2 & 6 \end{array}\right) $$

Determine whether the given matrix is orthogonal. $$ \left(\begin{array}{rrr} 0 & 0 & 1 \\ -12 & 5 & 0 \\ \frac{5}{13} & \frac{12}{13} & 0 \end{array}\right) $$

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

Suppose $$ \mathbf{A}=\left(\begin{array}{rrrr} 0 & 2 & 4 & 0 \\ 1 & 2 & -2 & 3 \\ 5 & 1 & 0 & -1 \\ 1 & 1 & 1 & 2 \end{array}\right) . $$ Evaluate the indicated minor determinant or cofactor. $$ C_{34} $$

State the appropriate theorem(s) in this section that justifies the given equality. Do not expand the determinants by cofactors. $$ \left|\begin{array}{rrrr} 0 & 5 & 0 & 6 \\ 2 & 1 & 0 & 8 \\ 0 & 2 & 0 & -9 \\ 0 & 6 & 0 & 4 \end{array}\right|=0 $$

Determine whether the given matrices are equal. $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right),\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right) $$

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

In Problems 1-20, fill in the blanks or answer true/false. If \(\mathbf{A}\) is a \(3 \times 3\) matrix such that \(\operatorname{det} \mathbf{A}=5\), then \(\operatorname{det}\left(\frac{1}{2} \mathbf{A}\right)=\) _________ and \(\operatorname{det}\left(-\mathbf{A}^{T}\right)=\) _________

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{ll} \frac{1}{2} & \frac{1}{6} \\ \frac{1}{6} & \frac{1}{2} \end{array}\right) $$

In Problems 7-10, use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{ll} 3 & 2 \\ 2 & 6 \end{array}\right) $$

$$ \text { In Problems } 5-10 \text {, determine whether the given matrix is orthogonal. } $$ $$ \left(\begin{array}{rrr} 0 & 0 & 1 \\ -\frac{12}{13} & \frac{5}{13} & 0 \\ \frac{5}{13} & \frac{12}{13} & 0 \end{array}\right) $$

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. $$ \left(\begin{array}{rr} -1 & 2 \\ -7 & 8 \end{array}\right) $$

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} x_{1}-2 x_{2}-3 x_{3} &=3 \\ x_{1}+x_{2}-x_{3} &=5 \\ 3 x_{1}+2 x_{2} &=-4 \end{aligned} $$

In Problems 5-8, suppose $$ \mathbf{A}=\left(\begin{array}{rrrr} 0 & 2 & 4 & 0 \\ 1 & 2 & -2 & 3 \\ 5 & 1 & 0 & -1 \\ 1 & 1 & 1 & 2 \end{array}\right) $$ \text { Evaluate the indicated minor determinant or cofactor. } $$ C_{34} $$

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

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right),\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right) $$

In an experiment the following correspondence was found between temperature \(T\) (in \({ }^{\circ} \mathrm{C}\) ) and electrical resistance \(R\) (in M\Omega): $$ \begin{array}{|lrrrrrr|} \hline \boldsymbol{T} & 400 & 450 & 500 & 550 & 600 & 650 \\ \hline R & 0.47 & 0.90 & 2.0 & 3.7 & 7.5 & 15 \\ \hline \end{array} $$ Find the least squares line \(R=a T+b\). Use this line toestimate the resistance at \(T=700\).

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rr} -2 & -1 \\ 1 & -4 \end{array}\right) $$

Use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{ll} 1 & 3 \\ 3 & 9 \end{array}\right) $$

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

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{rrr} 2 & 3 & 0 \\ 0 & 11 & 14 \\ -1 & 4 & 7 \end{array}\right) $$

Solve the given system of equations by Cramer's rule. $$ \begin{aligned} x_{1}-x_{2}+6 x_{3} &=-2 \\ -x_{1}+2 x_{2}+4 x_{3} &=9 \\ 2 x_{1}+3 x_{2}-x_{3} &=\frac{1}{2} \end{aligned} $$

Determine whether the given matrices are equal. $$ \left(\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right) $$

In Problems 1-20, fill in the blanks or answer true/false. $$ \text { If } \operatorname{det} \mathbf{A}=6 \text { and } \operatorname{det} \mathbf{B}=2 \text {, then } \operatorname{det} \mathbf{A} \mathbf{B}^{-1}= $$ _________

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} -2 & -1 \\ 1 & -4 \end{array}\right) $$

In Problems 7-10, use the method of deflation to find the eigenvalues of the given matrix. $$ \left(\begin{array}{ll} 1 & 3 \\ 3 & 9 \end{array}\right) $$

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

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. $$ \left(\begin{array}{ll} 2 & 1 \\ 2 & 1 \end{array}\right) $$

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} x_{1}-x_{2}+6 x_{3} &=-2 \\ -x_{1}+2 x_{2}+4 x_{3} &=9 \\ 2 x_{1}+3 x_{2}-x_{3} &=\frac{1}{2} \end{aligned} $$

In Problems 5-8, suppose $$ \mathbf{A}=\left(\begin{array}{rrrr} 0 & 2 & 4 & 0 \\ 1 & 2 & -2 & 3 \\ 5 & 1 & 0 & -1 \\ 1 & 1 & 1 & 2 \end{array}\right) $$ \text { Evaluate the indicated minor determinant or cofactor. } $$ C_{23} $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} x_{1}+2 x_{2}-4 x_{3}=9 \\ 5 x_{1}-x_{2}+2 x_{3}=1 \end{array} $$

In Problems 7-10, determine whether the given matrices are equal. $$ \left(\begin{array}{ll} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right) $$