Chapter 8

Suppose the system \(\mathbf{A} \mathbf{X}=\mathbf{B}\) is consistent and \(\mathbf{A}\) is a \(6 \times 3\) matrix. Suppose the maximum number of linearly independent rows in \(\mathbf{A}\) is 3 . Discuss: Is the solution of the system unique?

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(\begin{aligned} 2 x_{1}+x_{2}+x_{3} &=3 \\ 3 x_{1}+x_{2}+x_{3}+x_{4} &=4 \\\ x_{1}+2 x_{2}+2 x_{3}+3 x_{4} &=3 \\ 4 x_{1}+5 x_{2}-2 x_{3}+x_{4} &=16 \end{aligned}\)

In Problems 1-20, fill in the blanks or answer true/false. The only matrices that are orthogonally diagonalizable are symmetric matrices._________

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrr} 0 & 0 & 1 \\ 1 & 0 & -3 \\ 0 & 1 & 3 \end{array}\right) $$

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.) $$ \left(\begin{array}{rrr} 2 & 8 & -2 \\ 8 & -4 & 10 \\ -2 & 10 & -7 \end{array}\right) $$

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. $$ \left(\begin{array}{lll} 1 & 6 & 0 \\ 0 & 2 & 1 \\ 0 & 1 & 2 \end{array}\right) $$

In Problems 15-18, evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{D}=\left(\begin{array}{rrr} 0 & 7 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & -2 \end{array}\right) $$

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrr} 1 & -1 & -1 \\ 2 & 2 & -2 \\ 1 & 1 & 9 \end{array}\right) $$

Suppose the system \(\mathbf{A X}=\mathbf{B}\) is consistent and \(\mathbf{A}\) is a \(6 \times 3\) matrix. Suppose the maximum number of linearly independent rows in \(\mathbf{A}\) is 3 . Discuss: Is the solution of the system unique?

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} 2 x_{1}+x_{2}+x_{3} &=3 \\ 3 x_{1}+x_{2}+x_{3}+x_{4} &=4 \\ x_{1}+2 x_{2}+2 x_{3}+3 x_{4} &=3 \\ 4 x_{1}+5 x_{2}-2 x_{3}+x_{4} &=16 \end{aligned} $$

If \(\mathbf{A}=\left(\begin{array}{rr}1 & 4 \\ 5 & 10 \\ 8 & 12\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rrr}-4 & 6 & -3 \\\ 1 & -3 & 2\end{array}\right)\), find (a) \(\mathbf{A B}\), (b) \(\mathbf{B A}\).

Determine whether the given message is a code word in the Hamming \((7,4)\) code. If it is, decode it. If it is not, correct the single error and decode the corrected message. $$ \left.\begin{array}{llllllll} 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rrrr} -8 & -10 & 7 & -9 \\ 0 & 2 & 0 & 0 \\ -9 & -9 & 8 & -9 \\ 1 & 1 & -1 & 2 \end{array}\right) $$

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right) $$

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{rr} 1 & -2 \\ -2 & 4 \end{array}\right), \mathbf{B}=\left(\begin{array}{ll} 6 & 3 \\ 2 & 1 \end{array}\right), \text { and } \mathbf{C}=\left(\begin{array}{ll} 0 & 2 \\ 3 & 4 \end{array}\right) \\ &\text { find (a) } \mathbf{B C},(\mathbf{b}) \mathbf{A}(\mathbf{B C}),(\mathbf{c}) \mathbf{C}(\mathbf{B A}),(\mathbf{d}) \mathbf{A}(\mathbf{B}+\mathbf{C}) \end{aligned} $$

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} 4 & 5 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{array}\right) $$

Verify that \(\operatorname{det} \mathbf{A}=\operatorname{det} \mathbf{A}^{T}\) for the given matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & 2 & 1 \\ 4 & 1 & -1 \\ 1 & 2 & -1 \end{array}\right) $$

Suppose we wish to determine whether the set of column vectors $$ \begin{array}{r} \mathbf{v}_{1}=\left(\begin{array}{l} 4 \\ 3 \\ 2 \\ 1 \end{array}\right), \quad \mathbf{v}_{2}=\left(\begin{array}{l} 1 \\ 2 \\ 2 \\ 1 \end{array}\right), \quad \mathbf{v}_{3}=\left(\begin{array}{r} -1 \\ 1 \\ 1 \\ 1 \end{array}\right) \\ \mathbf{v}_{4}=\left(\begin{array}{l} 2 \\ 3 \\ 4 \\ 1 \end{array}\right), \quad \mathbf{v}_{5}=\left(\begin{array}{r} 1 \\ 7 \\ -5 \\ 1 \end{array}\right) \end{array} $$ is linearly dependent or linearly independent. By Definition \(7.6 .3\), if $$ c_{1} \mathbf{v}_{1}+c_{2} \mathbf{v}_{2}+c_{3} \mathbf{v}_{3}+c_{4} \mathbf{v}_{4}+c_{5} \mathbf{v}_{5}=\mathbf{0} $$ only for \(c_{1}=0, c_{2}=0, c_{3}=0, c_{4}=0, c_{5}=0\), then the set of vectors is linearly independent; otherwise the set is linearly dependent. But (4) is equivalent to the linear system $$ \begin{array}{r} 4 c_{1}+c_{2}-c_{3}+2 c_{4}+c_{5}=0 \\ 3 c_{1}+2 c_{2}+c_{3}+3 c_{4}+7 c_{5}=0 \\ 2 c_{1}+2 c_{2}+c_{3}+4 c_{4}-5 c_{5}=0 \\ c_{1}+c_{2}+c_{3}+c_{4}+c_{5}=0 \end{array} $$ Without doing any further work, explain why we can now conclude that the set of vectors is linearly dependent.

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

In Problems 19-28, determine whether the given message is a code word in the Hamming \((7,4)\) code. If it is, decode it. If it is not, correct the single error and decode the corrected message. $$ \left(\begin{array}{lllllll} 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrrr} -8 & -10 & 7 & -9 \\ 0 & 2 & 0 & 0 \\ -9 & -9 & 8 & -9 \\ 1 & 1 & -1 & 2 \end{array}\right) $$

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. $$ \left(\begin{array}{rrr} 0 & 0 & -1 \\ 1 & 0 & 0 \\ 1 & 1 & -1 \end{array}\right) $$

In Problems 19 and 20, verify that \(\operatorname{det} \mathbf{A}=\operatorname{det} \mathbf{A}^{T}\) for the given matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & 2 & 1 \\ 4 & 1 & -1 \\ 1 & 2 & -1 \end{array}\right) $$

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} 4 & 5 & 3 \\ 1 & 2 & 3 \\ 1 & 2 & 3 \end{array}\right) $$

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

If \(\mathbf{A}=\left(\begin{array}{rr}1 & -2 \\ -2 & 4\end{array}\right), \mathbf{B}=\left(\begin{array}{ll}6 & 3 \\ 2 & 1\end{array}\right)\), and \(\mathbf{C}=\left(\begin{array}{ll}0 & 2 \\ 3 & 4\end{array}\right)\) find (a) \(\mathbf{B C}\), (b) \(\mathbf{A}(\mathbf{B C})\), (c) \(\mathbf{C}(\mathbf{B} \mathbf{A})\), (d) \(\mathbf{A}(\mathbf{B}+\mathbf{C})\).

Determine whether the given message is a code word in the Hamming \((7,4)\) code. If it is, decode it. If it is not, correct the single error and decode the corrected message. $$ \left.\begin{array}{llllllll} (1 & 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right) $$

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\). $$ \left(\begin{array}{rrrr} 4 & 2 & -1 & 4 \\ 0 & 2 & 0 & 0 \\ 1 & 3 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right) $$

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{rrr} 1 & 0 & -1 \\ 0 & -2 & 1 \\ 2 & -1 & 3 \end{array}\right) $$

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{lll} 5 & -6 & 7 \end{array}\right), \mathbf{B}=\left(\begin{array}{r} 3 \\ 4 \\ -1 \end{array}\right)\\\ &\text { and } \mathbf{C}=\left(\begin{array}{rrr} 1 & 2 & 4 \\ 0 & 1 & -1 \\ 3 & 2 & 1 \end{array}\right), \text { find }(\mathbf{a}) \mathbf{A B},(\mathbf{b}) \mathbf{B A},(\mathbf{c})(\mathbf{B A}) \mathbf{C},\\\ &\text { (d) }(\mathbf{A B}) \mathbf{C} \text { . } \end{aligned} $$

Verify that \(\operatorname{det} \mathbf{A}=\operatorname{det} \mathbf{A}^{T}\) for the given matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & 0 & 5 \\ 7 & 2 & -1 \end{array}\right) $$

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} \frac{1}{4} & 6 & 0 \\ \frac{1}{3} & 8 & 0 \\ \frac{1}{2} & 9 & 0 \end{array}\right) $$

A CAS can be used to row reduce a matrix to a row-echelon form. Use a CAS to determine the ranks of the augmented matrix \((\mathbf{A} \mid \mathbf{B})\) and the coefficient matrix \(\mathbf{A}\) for $$ \begin{aligned} x_{1}+2 x_{2}-6 x_{3}+x_{4}+x_{5}+x_{6} &=2 \\ 5 x_{1}+2 x_{2}-2 x_{3}+5 x_{4}+4 x_{5}+2 x_{6} &=3 \\ 6 x_{1}+2 x_{2}-2 x_{3}+x_{4}+x_{5}+3 x_{6} &=-1 \\ -x_{1}+2 x_{2}+3 x_{3}+x_{4}-x_{5}+6 x_{6} &=0 \\ 9 x_{1}+7 x_{2}-2 x_{3}+x_{4}+4 x_{5} &=5 \end{aligned} $$ Is the system consistent or inconsistent? If consistent, solve the system.

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(x_{1}+2 x_{2}+\quad x_{4}=0\) \(4 x_{1}+9 x_{2}+x_{3}+12 x_{4}=0\) \(3 x_{1}+9 x_{2}+6 x_{3}+21 x_{4}=0\) \(x_{1}+3 x_{2}+x_{3}+9 x_{4}=0\)

In Problems 1-20, fill in the blanks or answer true/false. The eigenvalues of a symmetric matrix with real entries are always real numbers._________

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 \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A P}\). $$ \left(\begin{array}{rrrr} 4 & 2 & -1 & 4 \\ 0 & 2 & 0 & 0 \\ 1 & 3 & 2 & 1 \\ 0 & 0 & 0 & 2 \end{array}\right) $$

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. $$ \left(\begin{array}{rrr} 2 & -1 & 0 \\ 5 & 2 & 4 \\ 0 & 1 & 2 \end{array}\right) $$

In Problems 19 and 20, verify that \(\operatorname{det} \mathbf{A}=\operatorname{det} \mathbf{A}^{T}\) for the given matrix \(\mathbf{A}\). $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 4 \\ 1 & 0 & 5 \\ 7 & 2 & -1 \end{array}\right) $$

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} \frac{1}{4} & 6 & 0 \\ \frac{1}{3} & 8 & 0 \\ \frac{1}{2} & 9 & 0 \end{array}\right) $$

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} x_{1}+2 x_{2}+\quad x_{4} &=0 \\ 4 x_{1}+9 x_{2}+x_{3}+12 x_{4} &=0 \\ 3 x_{1}+9 x_{2}+6 x_{3}+21 x_{4} &=0 \\ x_{1}+3 x_{2}+x_{3}+9 x_{4} &=0 \end{aligned} $$

Determine whether the given message is a code word in the Hamming \((7,4)\) code. If it is, decode it. If it is not, correct the single error and decode the corrected message. $$ \left(\begin{array}{lllllll} 1 & 1 & 0 & 1 & 1 & 0 & 1 \end{array}\right) $$

(a) verify that the indicated column vectors are eigenvectors of the given symmetric matrix and (b) identify the corresponding eigenvalues. (c) Proceed as in Example 4 and use the Gram-Schmidt process to construct an orthogonal matrix \(\mathbf{P}\) from the eigenvectors. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{lll} 0 & 2 & 2 \\ 2 & 0 & 2 \\ 2 & 2 & 0 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right), \\ &\mathbf{K}_{2}=\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right) \end{aligned} $$

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. $$ \left(\begin{array}{rrr} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & -7 \end{array}\right) $$

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{rrr} 4 & 2 & 3 \\ 2 & 1 & 0 \\ -1 & -2 & 0 \end{array}\right) $$

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{r} 4 \\ 8 \\ -10 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{lll} 2 & 4 & 5 \end{array}\right), \text { find }(\mathbf{a}) \mathbf{A}^{T} \mathbf{A},(\mathbf{b}) \mathbf{B}^{T} \mathbf{B} \text { , }\\\ &\text { (c) } \mathbf{A}+\mathbf{B}^{T} \end{aligned} $$

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrr} -2 & -1 & 4 \\ -3 & 6 & 1 \\ -3 & 4 & 8 \end{array}\right) $$

Use a calculator to solve the given system. \(\begin{aligned} x_{1}+x_{2}+x_{3} &=4.280 \\ 0.2 x_{1}-0.1 x_{2}-0.5 x_{3} &=-1.978 \\ 4.1 x_{1}+0.3 x_{2}+0.12 x_{3} &=1.686 \end{aligned}\)

An \(n \times n\) matrix \(\mathbf{B}\) is symmetric if \(\mathbf{B}^{T}=\mathbf{B}\), and an \(n \times n\) matrix \(\mathbf{C}\) is skew- symmetric if \(\mathbf{C}^{T}=-\mathbf{C}\). By noting the identity \(2 \mathbf{A}=\mathbf{A}+\mathbf{A}^{T}+\mathbf{A}-\mathbf{A}^{T}\), show that any \(n \times n\) matrix \(\mathbf{A}\) can be written as the sum of a symmetric matrix and a skewsymmetric matrix.

In Problems 19-28, determine whether the given message is a code word in the Hamming \((7,4)\) code. If it is, decode it. If it is not, correct the single error and decode the corrected message. $$ \left(\begin{array}{lllllll} 1 & 1 & 0 & 1 & 1 & 0 & 1 \end{array}\right) $$

In Problems 21-30, the given matrix \(\mathbf{A}\) is symmetric. Find an orthogonal matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix D such that \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\). $$ \left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) $$