Chapter 8

Use the inverse power method to find the eigenvalue of least magnitude for the given matrix. $$ \left(\begin{array}{rr} -0.2 & 0.3 \\ 0.4 & -0.1 \end{array}\right) $$

To determine whether the given matrix is singular or nonsingular. $$ \left(\begin{array}{lll} 0 & 2 & 0 \\ 0 & 0 & 1 \\ 8 & 0 & 0 \end{array}\right) $$

Consider the system $$ \begin{aligned} &x_{1}+x_{2}=1 \\ &x_{1}+\varepsilon x_{2}=2 \end{aligned} $$ When \(\varepsilon\) is close to 1 , the lines that make up the system are 15 almost parallel. (a) Use Cramer's rule to show that a solution of the system is $$ x_{1}=1-\frac{1}{\varepsilon-1}, \quad x_{2}=\frac{1}{\varepsilon-1} $$ (b) The system is said to be ill-conditioned since small changes in the input data (for example, the coefficients) causes a significant or large change in the output or solution. Verify this by finding the solution of the system for \(\varepsilon=1.01\) and then for \(\varepsilon=0.99\).

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{lr} \frac{1}{4} & \frac{1}{2} \\ \frac{1}{3} & -\frac{4}{3} \end{array}\right) $$

Evaluate the determinant of the given matrix using the result \(\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\\ c_{1} & c_{2} & c_{3}\end{array}\right|=5\). $$ \mathbf{B}=\left(\begin{array}{rrr} 2 a_{1} & a_{2} & a_{3} \\ 6 b_{1} & 3 b_{2} & 3 b_{3} \\ 2 c_{1} & c_{2} & c_{3} \end{array}\right) $$

Determine the values of \(x\) and \(y\) for which the matrices are equal. $$ \left(\begin{array}{cc} x^{2} & 1 \\ y & 5 \end{array}\right),\left(\begin{array}{rr} 9 & 1 \\ 4 x & 5 \end{array}\right) $$

Determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 2,6,3\rangle, \mathbf{u}_{2}=\langle 1,-1,4\rangle, \mathbf{u}_{3}=\langle 3,2,1\rangle, \mathbf{u}_{4}=\langle 2,5,4\rangle $$

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(x_{1}-x_{2}-2 x_{3}=0\) \(2 x_{1}+4 x_{2}+5 x_{3}=0\) \(6 x_{1}-\quad 3 x_{3}=0\)

In Problems 1-20, fill in the blanks or answer true/false. A nonzero scalar multiple of an eigenvector is also an eigenvector corresponding to the same eigenvalue._________

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrr} 1 & 2 & 2 \\ 2 & 3 & -2 \\ -5 & 3 & 8 \end{array}\right) $$

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

In Problems 9-14, evaluate the determinant of the given matrix. $$ \left(\begin{array}{rr} \frac{1}{4} & \frac{1}{2} \\ \frac{1}{3} & -\frac{4}{3} \end{array}\right) $$

In Problems \(11-14\), determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 2,6,3\rangle, \mathbf{u}_{2}=\langle 1,-1,4\rangle, \mathbf{u}_{3}=\langle 3,2,1\rangle, \mathbf{u}_{4}=\langle 2,5,4\rangle $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} x_{1}-x_{2}-2 x_{3}=0 \\ 2 x_{1}+4 x_{2}+5 x_{3}=0 \\ 6 x_{1}-\quad 3 x_{3}=0 \end{array} $$

In Problems 11 and 12, determine the values of \(x\) and \(y\) for which the matrices are equal. $$ \left(\begin{array}{ll} x^{2} & 1 \\ y & 5 \end{array}\right),\left(\begin{array}{cc} 9 & 1 \\ 4 x & 5 \end{array}\right) $$

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

Show that \(\lambda=0\) is an eigenvalue of each matrix. In this case, the coefficient \(c_{0}\) in the characteristic equation (1) is 0 . Compute \(\mathbf{A}^{m}\) in each case. In parts (a) and (b), explain why we do not have to solve any system for the coefficients \(c_{i}\) in determining \(\mathbf{A}^{m}\). (a) \(\mathbf{A}=\left(\begin{array}{ll}1 & 1 \\ 3 & 3\end{array}\right)\) (b) \(\mathbf{A}=\left(\begin{array}{ll}1 & -1 \\ 1 & -1\end{array}\right)\) (c) \(\mathbf{A}=\left(\begin{array}{rrr}2 & 1 & 1 \\ 1 & 0 & -2 \\ 1 & 1 & 3\end{array}\right)\)

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

$$ \begin{aligned} &\text { In Problems } \underline{\phantom{xxx}} , \text { find the entries } c_{23} \text { and } c_{12} \text { for the matrix }\\\ &\mathbf{C}=2 \mathbf{A}-3 \mathbf{B} \end{aligned} $$ $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & -1 \\ -1 & 6 & 0 \end{array}\right), \mathbf{B}=\left(\begin{array}{rrr} 4 & -2 & 6 \\ 1 & 3 & -3 \end{array}\right) $$

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{cc} 1-\lambda & 3 \\ 2 & 2-\lambda \end{array}\right) $$

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

Determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 1,-1,3,-1\rangle, \mathbf{u}_{2}=\langle 1,-1,4,2\rangle, \mathbf{u}_{3}=\langle 1,-1,5,7\rangle $$

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

In Problems 1-20, fill in the blanks or answer true/false. An \(n \times 1\) column vector \(K\) with all zero entries is never an eigenvector of an \(n \times n\) matrix \(\mathbf{A}\)._________

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

In Problems 11-18, proceed as in Example 3 to construct an orthogonal matrix from the eigenvectors of the given symmetric matrix. (The answers are not unique.) $$ \left(\begin{array}{ll} 1 & 3 \\ 3 & 9 \end{array}\right) $$

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. $$ \left(\begin{array}{rr} 4 & 8 \\ 0 & -5 \end{array}\right) $$

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

In Problems 9-14, evaluate the determinant of the given matrix. $$ \left(\begin{array}{cc} 1-\lambda & 3 \\ 2 & 2-\lambda \end{array}\right) $$

In Problems \(11-14\), determine whether the given set of vectors is linearly dependent or linearly independent. $$ \mathbf{u}_{1}=\langle 1,-1,3,-1\rangle, \mathbf{u}_{2}=\langle 1,-1,4,2\rangle, \mathbf{u}_{3}=\langle 1,-1,5,7\rangle $$

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

In Problems 13 and 14, find the entries \(c_{23}\) and \(c_{12}\) for the matrix \(\mathbf{C}=2 \mathbf{A}-3 \mathbf{B}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & -1 \\ -1 & 6 & 0 \end{array}\right), \mathbf{B}=\left(\begin{array}{rrr} 4 & -2 & 6 \\ 1 & 3 & -3 \end{array}\right) $$

Encode the given word using the Hamming \((7,4)\) code. $$ \left(\begin{array}{lllllll} 0 & 0 & 1 & 1 \end{array}\right) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rrr} 0 & -9 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) $$

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

$$ \begin{aligned} &\text { In Problems } \underline{\phantom{xxx}} , \text { find the entries } c_{23} \text { and } c_{12} \text { for the matrix }\\\ &\mathbf{C}=2 \mathbf{A}-3 \mathbf{B} \end{aligned} $$ $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 0 & -4 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{lll} 2 & 0 & 5 \\ 0 & 4 & 0 \\ 3 & 0 & 7 \end{array}\right) $$

Evaluate the determinant of the given matrix. $$ \left(\begin{array}{cc} -3-\lambda & -4 \\ -2 & 5-\lambda \end{array}\right) $$

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

Determine whether the given set of vectors is linearly dependent or linearly independent. $$ \begin{aligned} &\mathbf{u}_{1}=\langle 2,1,1,5\rangle, \mathbf{u}_{2}=\langle 2,2,1,1\rangle, \mathbf{u}_{3}=\langle 3,-1,6,1\rangle, \\ &\mathbf{u}_{4}=\langle 1,1,1,-1\rangle \end{aligned} $$

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

In Problems 1-20, fill in the blanks or answer true/false. Let \(A\) be an \(n \times n\) matrix with real entries. If \(\lambda\) is a complex eigenvalue, then \(\bar{\lambda}\) is also an eigenvalue of \(\mathbf{A}\)._________

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrr} 0 & -9 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) $$

In Problems 11-18, proceed as in Example 3 to construct an orthogonal matrix from the eigenvectors of the given symmetric matrix. (The answers are not unique.) $$ \left(\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right) $$

In his work Liber Abbaci, published in 1202 , Leonardo Fibonacci of Pisa, Italy, speculated on the reproduction of rabbits: How many pairs of rabbits will be produced in a year beginning with a single pair, if every month each pair bears a new pair which become productive from the second month on? The answer to his question is contained in a sequence known as a Fibonacci sequence. $$ \begin{array}{|lccccccccccccc|} \hline \multicolumn{18}{|c|}{\text { After Each Month }} \\ \hline \text { Start } n= & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \text { Adult pairs } & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & \ldots & & & & \\ \text { Baby pairs } & 0 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & \ldots & & & & \\ \text { Total pairs } & 1 & 2 & 3 & 5 & 8 & 13 & 21 & 34 & \ldots & & & & \\ \hline \end{array} $$ Each of the three rows describing rabbit pairs is a Fibonacci sequence and can be defined recursively by a second-order difference equation \(x_{n}=x_{n-2}+x_{n-1}, n=2,3, \ldots\), where \(x_{0}\) and \(x_{1}\) depend on the row. For example, for the first row designating adult pairs of rabbits, \(x_{0}=1, x_{1}=1\). (a) If we let \(y_{n-1}=x_{n-2}\), then \(y_{n}=x_{n-1}\), and the difference equation can be written as a system of first-order difference equations $$ \begin{aligned} &x_{n}=x_{n-1}+y_{n-1} \\ &y_{n}=x_{n-1} . \end{aligned} $$ Write this system in the matrix form \(\mathbf{X}_{n}=\mathbf{A} \mathbf{X}_{n-1}\), \(n=2,3, \ldots\) (b) Show that $$ \mathbf{A}^{m}=\left(\begin{array}{cc} \frac{\lambda_{2} \lambda_{1}^{m}-\lambda_{1} \lambda_{2}^{m}+\lambda_{2}^{m}-\lambda_{1}^{m}}{\lambda_{2}-\lambda_{1}} & \frac{\lambda_{2}^{m}-\lambda_{1}^{m}}{\lambda_{2}-\lambda_{1}} \\ \frac{\lambda_{2}^{m}-\lambda_{1}^{m}}{\lambda_{2}-\lambda_{1}} & \frac{\lambda_{2} \lambda_{1}^{m}-\lambda_{1} \lambda_{2}^{m}}{\lambda_{2}-\lambda_{1}} \end{array}\right) $$ $$ \mathbf{A}^{m}=\frac{1}{2^{m+1} \sqrt{5}}\left(\begin{array}{cc} (1+\sqrt{5})^{m+1}-(1-\sqrt{5})^{m+1} & 2(1+\sqrt{5})^{m}-2(1-\sqrt{5})^{m} \\\ 2(1+\sqrt{5})^{m}-2(1-\sqrt{5})^{m} & (1+\sqrt{5})(1-\sqrt{5})^{m}-(1-\sqrt{5})(1+\sqrt{5})^{m} \end{array}\right) $$ where \(\lambda_{1}=\frac{1}{2}(1-\sqrt{5})\) and \(\lambda_{2}=\frac{1}{2}(1+\sqrt{5})\) are the distinct eigenvalues of \(\mathbf{A}\). (c) Use the result in part (a) to show \(\mathbf{X}_{n}=\mathbf{A}^{n-1} \mathbf{X}_{1}\). Use the last result and the result in part (b) to find thenumber of adult pairs, baby pairs, and total pairs of rabbits after the twelfth month.

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. $$ \left(\begin{array}{rr} 7 & 0 \\ 0 & 13 \end{array}\right) $$

In Problems 9-14, evaluate the determinant of the given matrix. $$ \left(\begin{array}{cc} -3-\lambda & -4 \\ -2 & 5-\lambda \end{array}\right) $$

In Problems \(11-14\), determine whether the given set of vectors is linearly dependent or linearly independent. $$ \begin{aligned} &\mathbf{u}_{1}=\langle 2,1,1,5\rangle, \mathbf{u}_{2}=\langle 2,2,1,1\rangle, \mathbf{u}_{3}=\langle 3,-1,6,1\rangle \\ &\mathbf{u}_{4}=\langle 1,1,1,-1\rangle \end{aligned} $$

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

In Problems 13 and 14, find the entries \(c_{23}\) and \(c_{12}\) for the matrix \(\mathbf{C}=2 \mathbf{A}-3 \mathbf{B}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & -1 & 1 \\ 2 & 2 & 1 \\ 0 & -4 & 1 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{lll} 2 & 0 & 5 \\ 0 & 4 & 0 \\ 3 & 0 & 7 \end{array}\right) $$

Encode the given word using the Hamming \((7,4)\) code. $$ \left(\begin{array}{lllll} 0 & 1 & 0 & 1 \end{array}\right) $$