Chapter 8

Find the least squares line for the given data. $$ (2,1),(3,2),(4,3),(5,2) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{ll} 2 & 3 \\ 1 & 4 \end{array}\right) $$

Verify that the given matrix satisfies its own characteristic equation. $$ \mathbf{A}=\left(\begin{array}{rr} 1 & -2 \\ 4 & 5 \end{array}\right) $$

In Problems, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A} .\) Give the corresponding eigenvalue. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{ll} 4 & 2 \\ 5 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} 5 \\ -2 \end{array}\right), \mathbf{K}_{2}=\left(\begin{array}{l} 2 \\ 5 \end{array}\right), \\ &\mathbf{K}_{3}=\left(\begin{array}{r} -2 \\ 5 \end{array}\right) \end{aligned} $$

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

Verify that the matrix \(\mathbf{B}\) is the inverse of the matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{ll} 1 & \frac{1}{2} \\ 2 & \frac{3}{2} \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr} 3 & -1 \\ -4 & 2 \end{array}\right) $$

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. $$ M_{12} $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(\begin{aligned} x_{1}-x_{2} &=11 \\ 4 x_{1}+3 x_{2} &=-5 \end{aligned}\)

State the size of the given matrix. $$ \left(\begin{array}{llll} 1 & 2 & 3 & 9 \\ 5 & 6 & 0 & 1 \end{array}\right) $$

In Problems 1-20, fill in the blanks or answer true/false. A matrix \(\mathbf{A}=\left(a_{i j}\right)_{4 \times 3}\) such that \(a_{i j}=i+j\) is given by _________.

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

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{ll} 2 & 3 \\ 1 & 4 \end{array}\right) $$

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. $$ \left(\begin{array}{rrr} 0 & 0 & -4 \\ 0 & -4 & 0 \\ -4 & 0 & 15 \end{array}\right) ; \quad\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 4 \\ 0 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ 0 \\ -4 \end{array}\right) $$

In Problems 1-6, determine which of the indicated column vectors are eigenvectors of the given matrix \(\mathbf{A}\). Give the corresponding eigenvalue. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{ll} 4 & 2 \\ 5 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} 5 \\ -2 \end{array}\right), \quad \mathbf{K}_{2}=\left(\begin{array}{l} 2 \\ 5 \end{array}\right) \\ &\mathbf{K}_{3}=\left(\begin{array}{r} -2 \\ 5 \end{array}\right) \end{aligned} $$

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

In Problems 1 and 2 , verify that the matrix \(\mathbf{B}\) is the inverse of the matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{ll} 1 & \frac{1}{2} \\ 2 & \frac{3}{2} \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rr} 3 & -1 \\ -4 & 2 \end{array}\right) $$

In Problems 1-10, state the appropriate theorem(s) in this section that justifies the given equality. Do not expand the determinants by cofactors. $$ \left|\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right|=-\left|\begin{array}{ll} 3 & 4 \\ 1 & 2 \end{array}\right| $$

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. $$ M_{12} $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} x_{1}-x_{2} &=11 \\ 4 x_{1}+3 x_{2} &=-5 \end{aligned} $$

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{llll} 1 & 2 & 3 & 9 \\ 5 & 6 & 0 & 1 \end{array}\right) $$

Find the least squares line for the given data. $$ (0,-1),(1,3),(2,5),(3,7) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rr} -4 & -5 \\ 8 & 10 \end{array}\right) $$

Use the power method as illustrated in Example 3 to find the dominant eigenvalue and a corresponding dominant eigenvector of the given matrix. $$ \left(\begin{array}{rr} -7 & 2 \\ 8 & -1 \end{array}\right) $$

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. $$ \left(\begin{array}{rrr} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{array}\right) ;\left(\begin{array}{r} -2 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 0 \\ 1 \\ -1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right) $$

Verify that the given matrix satisfies its own characteristic equation. $$ \mathbf{A}=\left(\begin{array}{lll} 0 & 1 & 2 \\ 1 & 0 & 3 \\ 0 & 1 & 1 \end{array}\right) $$

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

Solve the given system of equations by Cramer's rule. $$ \begin{array}{r} x_{1}+x_{2}=4 \\ 2 x_{1}-x_{2}=2 \end{array} $$

Verify that the matrix \(\mathbf{B}\) is the inverse of the matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -1 & 0 \\ 3 & 0 & 2 \\ 1 & 1 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrr} 2 & -1 & 2 \\ 1 & -1 & 2 \\ -3 & 2 & -3 \end{array}\right) $$

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. $$ M_{32} $$

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

State the size of the given matrix. $$ \left(\begin{array}{ll} 0 & 2 \\ 8 & 4 \\ 5 & 6 \end{array}\right) $$

In Problems 1-20, fill in the blanks or answer true/false. If \(\mathbf{A}\) is a \(4 \times 7\) matrix and \(\mathbf{B}\) is a \(7 \times 3\) matrix, then the size of \(A B\) is _________

In Problems \(1-6\), find the least squares line for the given data. $$ (0,-1),(1,3),(2,5),(3,7) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rr} -4 & -5 \\ 8 & 10 \end{array}\right) $$

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. $$ \left(\begin{array}{rrr} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{array}\right) ;\left(\begin{array}{r} -2 \\ 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 0 \\ 1 \\ -1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right) $$

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

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

In Problems 1 and 2 , verify that the matrix \(\mathbf{B}\) is the inverse of the matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -1 & 0 \\ 3 & 0 & 2 \\ 1 & 1 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{rrr} 2 & -1 & 2 \\ 1 & -1 & 2 \\ -3 & 2 & -3 \end{array}\right) $$

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. $$ M_{32} $$

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

In Problems \(1-6\), state the size of the given matrix. $$ \left(\begin{array}{ll} 0 & 2 \\ 8 & 4 \\ 5 & 6 \end{array}\right) $$

Find the least squares line for the given data. $$ (1,1),(2,1.5),(3,3),(4,4.5),(5,5) $$

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

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

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

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

Solve the given system of equations by Cramer's rule. $$ \begin{aligned} 0.1 x_{1}-0.4 x_{2} &=0.13 \\ x_{1}-\quad x_{2} &=0.4 \end{aligned} $$

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. $$ C_{13} $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(9 x_{1}+3 x_{2}=-5\) \(2 x_{1}+x_{2}=-1\)

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