Chapter 8

In Problems 21 and 22, (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 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. $$ \left(\begin{array}{rrr} 1 & 2 & 3 \\ 0 & 5 & 6 \\ 0 & 0 & -7 \end{array}\right) $$

Consider the matrices $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & -1 & 1 \\ 3 & 1 & -1 \\ 0 & 2 & 2 \end{array}\right) \quad \text { and } \quad \mathbf{B}=\left(\begin{array}{rrr} 2 & 1 & 5 \\ 4 & 3 & 8 \\ 0 & -1 & 0 \end{array}\right) $$ Verify that \(\operatorname{det} \mathbf{A B}=\operatorname{det} \mathbf{A} \cdot \operatorname{det} \mathbf{B}\).

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

In Problems 21 and 22 , use a calculator to solve the given system. $$ \begin{aligned} x_{1}+x_{2}+\quad x_{3} &=4.280 \\ 0.2 x_{1}-0.1 x_{2}-0.5 x_{3} &=-1.978 \\ 4.1 x_{1}+0.3 x_{2}+0.12 x_{3} &=1.686 \end{aligned} $$

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

In Problems, the given matrix \(\mathbf{A}\) is symmetric. Find an orthogonal matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\) $$ \left(\begin{array}{ll} 3 & 2 \\ 2 & 0 \end{array}\right) $$

(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. $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{llll} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{array}\right) ; \quad \mathbf{K}_{1}=\left(\begin{array}{r} -1 \\ 0 \\ 0 \\ 1 \end{array}\right), \\ &\mathbf{K}_{2}=\left(\begin{array}{r} -1 \\ 0 \\ 1 \\ 0 \end{array}\right), \quad \mathbf{K}_{3}=\left(\begin{array}{r} -1 \\ 1 \\ 0 \\ 0 \end{array}\right), \quad \mathbf{K}_{4}=\left(\begin{array}{l} 1 \\ 1 \\ 1 \\ 1 \end{array}\right) \end{aligned} $$

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

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{rrr} 2 & 4 & -2 \\ 4 & 2 & -2 \\ 8 & 10 & -6 \end{array}\right) $$

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{ll} 1 & 2 \\ 2 & 4 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{rr} -2 & 3 \\ 5 & 7 \end{array}\right), \text { find }\\\ &\text { (a) } \mathbf{A}+\mathbf{B}^{T} \text { , }\\\ &\text { (b) } 2 \mathbf{A}^{T}-\mathbf{B}^{T},(\mathbf{c}) \mathbf{A}^{T}(\mathbf{A}-\mathbf{B}) . \end{aligned} $$

Suppose \(\mathbf{A}\) is an \(n \times n\) matrix such that \(\mathbf{A}^{2}=\mathbf{I}\) where \(\mathbf{A}^{2}=\mathbf{A} \mathbf{A} .\) Show that \(\operatorname{det} \mathbf{A}=\pm 1\).

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrr} 3 & 5 & 1 \\ -1 & 2 & 5 \\ 7 & -4 & 10 \end{array}\right) $$

Use a calculator to solve the given system. \(\begin{aligned} 2.5 x_{1}+1.4 x_{2}+4.5 x_{3} &=2.6170 \\ 1.35 x_{1}+0.95 x_{2}+1.2 x_{3} &=0.7545 \\ 2.7 x_{1}+3.05 x_{2}-1.44 x_{3} &=-1.4292 \end{aligned}\)

Show that there exists no \(2 \times 2\) matrix with real entries such that \(\mathbf{A}^{2}=\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)\)

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

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 \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\). $$ \left(\begin{array}{ll} 3 & 2 \\ 2 & 0 \end{array}\right) $$

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

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrr} 3 & 5 & 1 \\ -1 & 2 & 5 \\ 7 & -4 & 10 \end{array}\right) $$

In Problems 21 and 22 , use a calculator to solve the given system. $$ \begin{aligned} 2.5 x_{1}+1.4 x_{2}+4.5 x_{3} &=2.6170 \\ 1.35 x_{1}+0.95 x_{2}+1.2 x_{3} &=0.7545 \\ 2.7 x_{1}+3.05 x_{2}-1.44 x_{3} &=-1.4292 \end{aligned} $$

If \(\mathbf{A}=\left(\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}-2 & 3 \\ 5 & 7\end{array}\right)\), find (a) \(\mathbf{A}+\mathbf{B}^{T}\) (b) \(2 \mathbf{A}^{T}-\mathbf{B}^{T}\),(c) \(\mathbf{A}^{T}(\mathbf{A}-\mathbf{B})\).

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

In Problems, the given matrix \(\mathbf{A}\) is symmetric. Find an orthogonal matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\) $$ \left(\begin{array}{rr} 5 & \sqrt{10} \\ \sqrt{10} & 8 \end{array}\right) $$

In Problems, find the eigenvalues and eigenvectors of the given nonsingular matrix \(\mathbf{A} .\) Then without finding \(\mathbf{A}^{-1}\), find its eigenvalues and corresponding eigenvectors. $$ \mathbf{A}=\left(\begin{array}{ll} 5 & 1 \\ 1 & 5 \end{array}\right) $$

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

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{ll} 3 & 4 \\ 8 & 1 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{rr} 5 & 10 \\ -2 & -5 \end{array}\right), \text { find }(\mathbf{a})(\mathbf{A B})^{T} \text { , }\\\ &\text { (b) } \mathbf{B}^{T} \mathbf{A}^{T} \end{aligned} $$

Consider the matrix $$ \mathbf{A}=\left(\begin{array}{lll} a & a+1 & a+2 \\ b & b+1 & b+2 \\ c & c+1 & c+2 \end{array}\right). $$ Without expanding, evaluate det \(\mathbf{A}\).

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

An \(n \times n\) matrix A is said to be nilpotent if, for some positive integer \(m, \mathbf{A}^{m}=\mathbf{0}\). Find a \(2 \times 2\) nilpotent matrix \(\mathbf{A} \neq \mathbf{0}\).

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

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

In Problems 23-26, find the eigenvalues and eigenvectors of the given nonsingular matrix \(\mathbf{A}\). Then without finding \(\mathbf{A}^{-1}\), find its eigenvalues and corresponding eigenvectors. $$ A=\left(\begin{array}{ll} 5 & 1 \\ 1 & 5 \end{array}\right) $$

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

In Problems 13 and 14, find the entries \(c_{23}\) and \(c_{12}\) for the matrix \(\mathbf{C}=2 \mathbf{A}-3 \mathbf{B}\). If \(\mathbf{A}=\left(\begin{array}{ll}3 & 4 \\ 8 & 1\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}5 & 10 \\ -2 & -5\end{array}\right)\), find (a) \((\mathbf{A B})^{T}\) (b) \(\mathbf{B}^{T} \mathbf{A}^{T}\).

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

In Problems, the given matrix \(\mathbf{A}\) is symmetric. Find an orthogonal matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\) $$ \left(\begin{array}{rr} 1 & -2 \\ -2 & 1 \end{array}\right) $$

Construct an orthogonal matrix from the eigenvectors of $$ \mathbf{A}=\left(\begin{array}{llll} 1 & 2 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 2 & 1 \end{array}\right) $$

In Problems, find the eigenvalues and eigenvectors of the given nonsingular matrix \(\mathbf{A} .\) Then without finding \(\mathbf{A}^{-1}\), find its eigenvalues and corresponding eigenvectors. $$ A=\left(\begin{array}{rr} 4 & 2 \\ 7 & -1 \end{array}\right) $$

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{lll} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 8 \end{array}\right) $$

Write the given sum as a single-column matrix. $$ 4\left(\begin{array}{r} -1 \\ 2 \end{array}\right)-2\left(\begin{array}{l} 2 \\ 8 \end{array}\right)+3\left(\begin{array}{r} -2 \\ 3 \end{array}\right) $$

Consider the matrix $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & 1 & 1 \\ x & y & z \\ y+z & x+z & x+y \end{array}\right). $$ Without expanding, show that det \(\mathbf{A}=0\).

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{ccc} 1 & 1 & 1 \\ x & y & z \\ 2+x & 3+y & 4+z \end{array}\right) $$

(a) Two \(n \times n\) matrices \(\mathbf{A}\) and \(\mathbf{B}\) are said to anticommute if \(\mathbf{A B}=-\mathbf{B A}\). Show that each of the Pauli spin matrices $$ \sigma_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \sigma_{y}=\left(\begin{array}{rr} 0 & -i \\ i & 0 \end{array}\right) \quad \sigma_{z}=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) $$ where \(i^{2}=-1\), anticommutes with the others. Pauli spin matrices are used in quantum mechanics. (b) The matrix \(\mathbf{C}=\mathbf{A B}-\mathbf{B A}\) is said to be the commutator of the \(n \times n\) matrices \(\mathbf{A}\) and \(\mathbf{B}\). Find the commutators of \(\sigma_{x}\) and \(\sigma_{y}, \sigma_{y}\) and \(\sigma_{z}\), and \(\sigma_{z}\) and \(\sigma_{x}\).

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

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

In Problems 23-26, find the eigenvalues and eigenvectors of the given nonsingular matrix \(\mathbf{A}\). Then without finding \(\mathbf{A}^{-1}\), find its eigenvalues and corresponding eigenvectors. $$ \mathbf{A}=\left(\begin{array}{rr} 4 & 2 \\ 7 & -1 \end{array}\right) $$

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

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

In Problems, the given matrix \(\mathbf{A}\) is symmetric. Find an orthogonal matrix \(\mathbf{P}\) that diagonalizes \(\mathbf{A}\) and the diagonal matrix \(\mathbf{D}\) such that \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\) $$ \left(\begin{array}{lll} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right) $$

Show that if \(\mathbf{A}\) and \(\mathbf{B}\) are \(n \times n\) orthogonal matrices, then \(\mathbf{A B}\) is orthogonal.