Chapter 8

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. $$ \mathbf{A}=\left(\begin{array}{rrr} 4 & 2 & -1 \\ 0 & 3 & -2 \\ 0 & 0 & 5 \end{array}\right) $$

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

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrrr} 1 & 1 & -3 & 0 \\ 1 & 5 & 3 & 2 \\ 1 & -2 & 1 & 0 \\ 4 & 8 & 0 & 0 \end{array}\right) $$

In Problems 25 and 26, solve the given system of equations by Gauss-Jordan elimination. $$ \left(\begin{array}{rrr} 5 & -1 & 1 \\ 2 & 4 & 0 \\ 1 & 1 & 5 \end{array}\right) \mathbf{X}=\left(\begin{array}{r} -9 \\ 27 \\ 9 \end{array}\right) $$

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

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

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. $$ \mathbf{A}=\left(\begin{array}{rrr} 4 & 2 & -1 \\ 0 & 3 & -2 \\ 0 & 0 & 5 \end{array}\right) $$

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrrr} 1 & 1 & -3 & 0 \\ 1 & 5 & 3 & 2 \\ 1 & -2 & 1 & 0 \\ 4 & 8 & 0 & 0 \end{array}\right) $$

In Problems 25-28, write the given sum as a single-column matrix. $$ 4\left(\begin{array}{r} -1 \\ 2 \end{array}\right)-2\left(\begin{array}{l} 2 \\ 8 \end{array}\right)+3\left(\begin{array}{r} -2 \\ 3 \end{array}\right) $$

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

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

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

Write the given sum as a single-column matrix. $$ \left(\begin{array}{r} 2 \\ 1 \\ -1 \end{array}\right)+5\left(\begin{array}{r} -1 \\ -1 \\ 3 \end{array}\right)-2\left(\begin{array}{r} 3 \\ 4 \\ -5 \end{array}\right) $$

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrrr} 2 & 1 & -2 & 1 \\ 0 & 5 & 0 & 4 \\ 1 & 6 & 1 & 0 \\ 5 & -1 & 1 & 1 \end{array}\right) $$

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

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

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

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrrr} 2 & 1 & -2 & 1 \\ 0 & 5 & 0 & 4 \\ 1 & 6 & 1 & 0 \\ 5 & -1 & 1 & 1 \end{array}\right) $$

In Problems 25-28, write the given sum as a single-column matrix. $$ 3\left(\begin{array}{r} 2 \\ 1 \\ -1 \end{array}\right)+5\left(\begin{array}{r} -1 \\ -1 \\ 3 \end{array}\right)-2\left(\begin{array}{r} 3 \\ 4 \\ -5 \end{array}\right) $$

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

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 \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\) $$ \left(\begin{array}{rrr} 5 & -2 & 0 \\ -2 & 6 & -2 \\ 0 & -2 & 7 \end{array}\right) $$

Use the given matrices to find \((\mathbf{A B})^{-1}\). $$ \mathbf{A}^{-1}=\left(\begin{array}{rr} 1 & -\frac{5}{2} \\ -\frac{1}{2} & \frac{3}{2} \end{array}\right), \quad \mathbf{B}^{-1}=\left(\begin{array}{rr} 2 & \frac{4}{3} \\ -\frac{1}{3} & \frac{5}{2} \end{array}\right) $$

Write the given sum as a single-column matrix. $$ \left(\begin{array}{rr} 2 & -3 \\ 1 & 4 \end{array}\right)\left(\begin{array}{r} -2 \\ 5 \end{array}\right)-\left(\begin{array}{rr} -1 & 6 \\ -2 & 3 \end{array}\right)\left(\begin{array}{r} -7 \\ 2 \end{array}\right) $$

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrrrr} 3 & 2 & 0 & 1 & -1 \\ 0 & 1 & 4 & 2 & 3 \\ 0 & 0 & 2 & -1 & 1 \\ 0 & 0 & 0 & 4 & 3 \\ 0 & 0 & 0 & 0 & 2 \end{array}\right) $$

Without expanding, show that \(\left|\begin{array}{ccc}1 & 1 & 1 \\\ \frac{1}{a} & \frac{1}{b} & \frac{1}{c} \\ b c & a c & a b\end{array}\right|=0\)

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

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 \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\). $$ \left(\begin{array}{rrr} 5 & -2 & 0 \\ -2 & 6 & -2 \\ 0 & -2 & 7 \end{array}\right) $$

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

In Problems 25-28, write the given sum as a single-column matrix. $$ \left(\begin{array}{rr} 2 & -3 \\ 1 & 4 \end{array}\right)\left(\begin{array}{r} -2 \\ 5 \end{array}\right)-\left(\begin{array}{rr} -1 & 6 \\ -2 & 3 \end{array}\right)\left(\begin{array}{r} -7 \\ 2 \end{array}\right) $$

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

True or false: If \(\lambda\) is an eigenvalue of an \(n \times n\) matrix \(\mathbf{A}\), then the matrix \(\mathbf{A}-\lambda \mathbf{I}\) is singular. Justify your answer.

Write the given sum as a single-column matrix. $$ \left(\begin{array}{rrr} 1 & -3 & 4 \\ 2 & 5 & -1 \\ 0 & -4 & -2 \end{array}\right)\left(\begin{array}{r} 3 \\ 2 \\ -1 \end{array}\right)+\left(\begin{array}{r} -1 \\ 1 \\ 4 \end{array}\right)-\left(\begin{array}{r} 2 \\ 8 \\ -6 \end{array}\right) $$

Use the given matrices to find \((\mathbf{A B})^{-1}\). $$ \mathbf{A}^{-1}=\left(\begin{array}{rrr} 1 & 3 & -15 \\ 0 & -1 & 5 \\ -1 & -2 & 11 \end{array}\right), \quad \mathbf{B}^{-1}=\left(\begin{array}{rrr} -1 & 1 & 0 \\ 2 & 0 & 0 \\ 1 & 1 & -2 \end{array}\right) $$

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrrrr} 2 & 2 & 0 & 0 & -2 \\ 1 & 1 & 6 & 0 & 5 \\ 1 & 0 & 2 & -1 & -1 \\ 2 & 0 & 1 & -2 & 3 \\ 0 & 1 & 0 & 0 & 1 \end{array}\right) $$

Show that \(\left|\begin{array}{cccc}y & x^{2} & x & 1 \\ 2 & 1 & 1 & 1 \\ 3 & 4 & 2 & 1 \\ 5 & 9 & 3 & 1\end{array}\right|=0\) is the equation of a parabola passing through the three points \((1,2),(2,3)\), and \((3,5)\).

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

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

In Problems 27 and 28 , use the given matrices to find \((\mathbf{A B})^{-1}\). $$ \mathbf{A}^{-1}=\left(\begin{array}{rrr} 1 & 3 & -15 \\ 0 & -1 & 5 \\ -1 & -2 & 11 \end{array}\right), \quad \mathbf{B}^{-1}=\left(\begin{array}{rrr} -1 & 1 & 0 \\ 2 & 0 & 0 \\ 1 & 1 & -2 \end{array}\right) $$

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrrrr} 2 & 2 & 0 & 0 & -2 \\ 1 & 1 & 6 & 0 & 5 \\ 1 & 0 & 2 & -1 & -1 \\ 2 & 0 & 1 & -2 & 3 \\ 0 & 1 & 0 & 0 & 1 \end{array}\right) $$

In Problems 25-28, write the given sum as a single-column matrix. $$ \left(\begin{array}{rrr} 1 & -3 & 4 \\ 2 & 5 & -1 \\ 0 & -4 & -2 \end{array}\right)\left(\begin{array}{r} 3 \\ 2 \\ -1 \end{array}\right)+\left(\begin{array}{r} -1 \\ 1 \\ 4 \end{array}\right)-\left(\begin{array}{r} 2 \\ 8 \\ -6 \end{array}\right) $$

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 \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\) $$ \left(\begin{array}{lll} 1 & 0 & 7 \\ 0 & 1 & 0 \\ 7 & 0 & 1 \end{array}\right) $$

Determine the size of the matrix \(\mathbf{A}\) such that the given product is defined. $$ \left(\begin{array}{llll} 2 & 1 & 3 & 3 \\ 9 & 6 & 7 & 0 \end{array}\right) \mathbf{A}\left(\begin{array}{l} 0 \\ 5 \\ 7 \\ 9 \\ 2 \end{array}\right) $$

If \(\mathbf{A}^{-1}=\left(\begin{array}{ll}4 & 3 \\ 3 & 2\end{array}\right)\), what is \(\mathbf{A}\) ?

Find the values of \(\lambda\) that satisfy the given equation. $$ \left|\begin{array}{cc} -3-\lambda & 10 \\ 2 & 5-\lambda \end{array}\right|=0 $$

In Problems 29 and 30, evaluate the determinant of the given matrix by inspection. $$ \left(\begin{array}{rrrrrr} 4 & 0 & 0 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 \end{array}\right) $$

$$ \left(\begin{array}{rrr} 1 & -2 & 1 \\ 0 & 1 & 2 \\ 2 & 6 & 1 \end{array}\right)\left(\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right)=\left(\begin{array}{r} 14 \\ -42 \\ 7 \end{array}\right) $$

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 \(\mathbf{D}=\mathbf{P}^{T} \mathbf{A P}\). $$ \left(\begin{array}{lll} 1 & 0 & 7 \\ 0 & 1 & 0 \\ 7 & 0 & 1 \end{array}\right) $$

$$ \text { If } \mathbf{A}^{-1}=\left(\begin{array}{ll} 4 & 3 \\ 3 & 2 \end{array}\right) \text {, what is } \mathbf{A} \text { ? } $$

(a) In the Hamming \((8,4)\) code a word $$ \mathbf{W}=\left(w_{1} w_{2} w_{3} w_{4}\right) $$ of length four is transformed into a code word of length eight: $$ \mathbf{C}=\left(c_{1} c_{2} c_{3} w_{1} c_{4} w_{2} w_{3} w_{4}\right), $$ where the parity check equations are $$ \begin{aligned} &c_{4}+w_{2}+w_{3}+w_{4}=0 \\ &c_{3}+w_{1}+w_{3}+w_{4}=0 \\ &c_{2}+w_{1}+w_{2}+w_{4}=0 \\ &c_{1}+c_{2}+c_{3}+w_{1}+c_{4}+w_{2}+w_{3}+w_{4}=0. \end{aligned} $$ Encode the word \(\left(\begin{array}{lll}0 & 1 & 1 & 0\end{array}\right)\). (b) From the system in part (a), determine the parity check matrix \(\mathbf{H}\). (c) Using the matrix \(\mathbf{H}\) from part (b), compute the syndrome S of the received message $$ \mathbf{R}=\left(\begin{array}{lllllll} 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \end{array}\right) . $$