Chapter 8

In Problems 1-20, fill in the blanks or answer true/false. If \(\mathbf{A}=\left(\begin{array}{l}1 \\ 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{ll}3 & 4\end{array}\right)\), then \(\mathbf{A B}=\) _________ and \(\mathbf{B} \mathbf{A}=\)_________.

In Problems \(1-6\), find the least squares line for the given data. $$ (1,1),(2,1.5),(3,3),(4,4.5),(5,5) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} 0 & 1 \\ -1 & 2 \end{array}\right) $$

In Problems 3-6, use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. $$ \left(\begin{array}{rr} 2 & 4 \\ 3 & 13 \end{array}\right) $$

In Problems \(1-4\), (a) verify that the indicated column vectors are eigenvectors of the given symmetric matrix, (b) identify the corresponding eigenvalues, and (c) verify that the column vectors are orthogonal. $$ \begin{aligned} &\left(\begin{array}{rrr} 5 & 13 & 0 \\ 13 & 5 & 0 \\ 0 & 0 & -8 \end{array}\right)\\\ &\left(\begin{array}{l} \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \\ 0 \end{array}\right),\left(\begin{array}{c} \frac{\sqrt{3}}{3} \\ -\frac{\sqrt{3}}{3} \\ \frac{\sqrt{3}}{3} \end{array}\right), \quad\left(\begin{array}{c} \frac{\sqrt{6}}{6} \\ -\frac{\sqrt{6}}{6} \\ -\frac{\sqrt{6}}{3} \end{array}\right) \end{aligned} $$

In Problems 1-6, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A}\). Give the corresponding eigenvalue. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{ll} 0 & 5 \\ 2 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} 5 \\ -2 \end{array}\right) \\ &\mathbf{K}_{2}=\left(\begin{array}{l} 1 \\ 0 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{c} -5 \\ 10 \end{array}\right) \end{aligned} $$

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} 0.1 x_{1}-0.4 x_{2} &=0.13 \\ x_{1}-x_{2} &=0.4 \end{aligned} $$

In Problems \(1-4\), suppose $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & -1 & 2 \\ -2 & 3 & 5 \end{array}\right) $$ Evaluate the indicated minor determinant or cofactor. $$ C_{13} $$

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

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

Find the least squares line for the given data. $$ (0,0),(2,1.5),(3,3),(4,4.5),(5,5) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{ll} 0 & 5 \\ 1 & 0 \end{array}\right) $$

Use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. $$ \left(\begin{array}{ll} -1 & 2 \\ -2 & 7 \end{array}\right) $$

In Problems, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A} .\) Give the corresponding eigenvalue. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rr} 2 & 8 \\ -1 & -2 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{l} 0 \\ 0 \end{array}\right), \\ &\mathbf{K}_{2}=\left(\begin{array}{c} 2+2 i \\ -1 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{c} 2+2 i \\ 1 \end{array}\right) \end{aligned} $$

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

Solve the given system of equations by Cramer's rule. $$ \begin{array}{r} 0.21 x_{1}+0.57 x_{2}=0.369 \\ 0.1 x_{1}+0.2 x_{2}=0.135 \end{array} $$

Suppose $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & -1 & 2 \\ -2 & 3 & 5 \end{array}\right) $$ Evaluate the indicated minor determinant or cofactor. $$ C_{22} $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(10 x_{1}+15 x_{2}=1\) \(3 x_{1}+2 x_{2}=-1\)

State the size of the given matrix. $$ \left(\begin{array}{lll} 5 & 7 & -15 \end{array}\right) $$

In Problems 1-20, fill in the blanks or answer true/false. $$ \text { If } \mathbf{A}=\left(\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right), \text { then } \mathbf{A}^{-1}= $$ _________.

In Problems \(1-6\), find the least squares line for the given data. $$ (0,0),(2,1.5),(3,3),(4,4.5),(5,5) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{ll} 0 & 5 \\ 1 & 0 \end{array}\right) $$

In Problems \(1-4\), (a) verify that the indicated column vectors are eigenvectors of the given symmetric matrix, (b) identify the corresponding eigenvalues, and (c) verify that the column vectors are orthogonal. $$ \left(\begin{array}{lll} 3 & 2 & 2 \\ 2 & 2 & 0 \\ 2 & 0 & 4 \end{array}\right) ;\left(\begin{array}{r} -2 \\ 2 \\ 1 \end{array}\right), \quad\left(\begin{array}{r} 1 \\ 2 \\ -2 \end{array}\right),\left(\begin{array}{l} 2 \\ 1 \\ 2 \end{array}\right) $$

In Problems 1-6, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A}\). Give the corresponding eigenvalue. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{rr} 2 & 8 \\ -1 & -2 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{l} 0 \\ 0 \end{array}\right) \\ &\mathbf{K}_{2}=\left(\begin{array}{c} 2+2 i \\ -1 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{c} 2+2 i \\ 1 \end{array}\right) \end{aligned} $$

In Problems \(1-10\), solve the given system of equations by Cramer's rule. $$ \begin{aligned} 0.21 x_{1}+0.57 x_{2} &=0.369 \\ 0.1 x_{1}+0.2 x_{2} &=0.135 \end{aligned} $$

In Problems \(1-4\), suppose $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & -1 & 2 \\ -2 & 3 & 5 \end{array}\right) $$ Evaluate the indicated minor determinant or cofactor. $$ C_{22} $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} 10 x_{1}+15 x_{2} &=1 \\ 3 x_{1}+2 x_{2} &=-1 \end{aligned} $$

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{lll} 5 & 7 & -15 \end{array}\right) $$

Find the least squares line for the given data. $$ (0,2),(1,3),(2,5),(3,5),(4,9),(5,8),(6,10) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rr} -9 & 13 \\ -2 & 6 \end{array}\right) $$

Use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. $$ \left(\begin{array}{lll} 5 & 4 & 2 \\ 4 & 5 & 2 \\ 2 & 2 & 2 \end{array}\right) $$

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

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

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

Solve the given system of equations by Cramer's rule. $$ \begin{aligned} &2 x+y=1 \\ &3 x+2 y=-2 \end{aligned} $$

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. $$ M_{33} $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(\begin{aligned} x_{1}-x_{2}-x_{3} &=-3 \\ 2 x_{1}+3 x_{2}+5 x_{3} &=7 \\\ x_{1}-2 x_{2}+3 x_{3} &=-11 \end{aligned}\)

State the size of the given matrix. $$ \left(\begin{array}{rrrr} 1 & 5 & -6 & 0 \\ 7 & -10 & 2 & 12 \\ 0 & 9 & 2 & -1 \end{array}\right) $$

In Problems \(1-6\), find the least squares line for the given data. $$ (0,2),(1,3),(2,5),(3,5),(4,9),(5,8),(6,10) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} -9 & 13 \\ -2 & 6 \end{array}\right) $$

In Problems 3-6, use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. $$ \left(\begin{array}{lll} 5 & 4 & 2 \\ 4 & 5 & 2 \\ 2 & 2 & 2 \end{array}\right) $$

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

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

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

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. } $$ M_{33} $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} x_{1}-x_{2}-x_{3} &=-3 \\ 2 x_{1}+3 x_{2}+5 x_{3} &=7 \\ x_{1}-2 x_{2}+3 x_{3} &=-11 \end{aligned} $$

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{rrrr} 1 & 5 & -6 & 0 \\ 7 & -10 & 2 & 12 \\ 0 & 9 & 2 & -1 \end{array}\right) $$

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

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