Chapter 8

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

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{rr} 6 & -2 \\ 0 & 4 \end{array}\right) $$

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{rr} 4 & 5 \\ -6 & 9 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{rr} -2 & 6 \\ 8 & -10 \end{array}\right), \text { find (a) } \mathbf{A}+\mathbf{B}\\\ &\text { (b) } \mathbf{B}-\mathbf{A},(\mathbf{c}) 2 \mathbf{A}+3 \mathbf{B} \end{aligned} $$

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

Evaluate the determinant of the given matrix without expanding by cofactors. $$ A=\left(\begin{array}{rrrr} 6 & 1 & 8 & 10 \\ 0 & 2 & 7 & 2 \\ 0 & 0 & -4 & 9 \\ 0 & 0 & 0 & -5 \end{array}\right) $$

Suppose the system \(\mathbf{A X}=\mathbf{B}\) is consistent and \(\mathbf{A}\) is a \(5 \times 8\) matrix and \(\operatorname{rank}(\mathbf{A})=3 .\) How many parameters does the solution of the system have?

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

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

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

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. $$ \left(\begin{array}{rrr} 5 & -1 & 0 \\ 0 & -5 & 9 \\ 5 & -1 & 0 \end{array}\right) $$

In Problems 15-18, evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{A}=\left(\begin{array}{rrrr} 6 & 1 & 8 & 10 \\ 0 & \frac{2}{3} & 7 & 2 \\ 0 & 0 & -4 & 9 \\ 0 & 0 & 0 & -5 \end{array}\right) $$

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

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

If \(\mathbf{A}=\left(\begin{array}{rr}4 & 5 \\ -6 & 9\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}-2 & 6 \\ 8 & -10\end{array}\right)\), find (a) \(\mathbf{A}+\mathbf{B}\) (b) \(\mathbf{B}-\mathbf{A}\), (c) \(2 \mathbf{A}+3 \mathbf{B}\)

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

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

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. $$ \left(\begin{array}{lll} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 4 & 0 & 1 \end{array}\right) $$

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

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{rr} -2 & 0 \\ 4 & 1 \\ 7 & 3 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{rr} 3 & -1 \\ 0 & 2 \\ -4 & -2 \end{array}\right), \text { find (a) } \mathbf{A}-\mathbf{B}\\\ &\text { (b) } \mathbf{B}-\mathbf{A},(\mathbf{c}) 2(\mathbf{A}+\mathbf{B}) . \end{aligned} $$

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

Evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{B}=\left(\begin{array}{rrr} 0 & 0 & a_{13} \\ 0 & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) $$

Let \(\mathbf{A}\) be a nonzero \(4 \times 6\) matrix. (a) What is the maximum rank that \(\mathbf{A}\) can have? (b) If \(\operatorname{rank}(\mathbf{A} \mid \mathbf{B})=2\), then for what value(s) \(\operatorname{sen} \mathbf{r a n}(\mathbf{A})\) is the system \(\mathbf{A} \mathbf{X}=\mathbf{B}, \mathbf{B} \neq \mathbf{0}\), inconsistent? Consistent? (c) If \(\operatorname{rank}(\mathbf{A})=3\), then how many parameters does the solution of the system \(\mathbf{A} \mathbf{X}=\mathbf{0}\) have?

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(\begin{aligned} x_{1}-x_{2}-2 x_{3} &=-1 \\\\-3 x_{1}-2 x_{2}+x_{3} &=-7 \\\ 2 x_{1}+3 x_{2}+x_{3} &=8 \end{aligned}\)

In Problems 1-20, fill in the blanks or answer true/false. The augmented matrix $$ \left(\begin{array}{lll|l} 1 & 1 & 1 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right) $$ is in reduced rowechelon form._________

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

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. $$ \left(\begin{array}{lll} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 4 & 0 & 1 \end{array}\right) $$

In Problems 15-18, evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{B}=\left(\begin{array}{rrr} 0 & 0 & a_{13} \\ 0 & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right) $$

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

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{aligned} x_{1}-x_{2}-2 x_{3} &=-1 \\ -3 x_{1}-2 x_{2}+x_{3} &=-7 \\ 2 x_{1}+3 x_{2}+x_{3} &=8 \end{aligned} $$

If \(\mathbf{A}=\left(\begin{array}{rr}-2 & 0 \\ 4 & 1 \\ 7 & 3\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}3 & -1 \\ 0 & 2 \\\ -4 & -2\end{array}\right)\), find (a) \(\mathbf{A}-\mathbf{B}\) (b) \(\mathbf{B}-\mathbf{A}\), (c) \(2(\mathbf{A}+\mathbf{B})\)

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

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

A nonzero \(n \times n\) matrix \(\mathbf{A}\) is said to be nilpotent of index \(m\) if \(m\) is the smallest positive integer for which \(\mathbf{A}^{m}=\mathbf{0}\). Which of the following matrices are nilpotent? If nilpotent, what is its index? (a) \(\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right)\) (b) \(\left(\begin{array}{rr}2 & 2 \\ -2 & -2\end{array}\right)\) (c) \(\left(\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 3 & 0\end{array}\right)\) (d) \(\left(\begin{array}{lll}0 & 0 & 5 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)\) (e) \(\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ -1 & 0 & 0 & 0\end{array}\right)\) (f) \(\left(\begin{array}{llll}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0\end{array}\right)\)

Find the inverse of the given matrix or show that no inverse exists. $$ \left(\begin{array}{ll} 1 & 3 \\ 5 & 3 \end{array}\right) $$

If \(\mathbf{A}=\left(\begin{array}{rr}2 & -3 \\ -5 & 4\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}-1 & 6 \\ 3 & 2\end{array}\right)\), find \((\mathbf{a}) \mathbf{A B}\) (b) \(\mathbf{B A},(\mathbf{c}) \mathbf{A}^{2}=\mathbf{A} \mathbf{A}\) (d) \(\mathbf{B}^{2}=\mathbf{B B}\)

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} 3 & 0 & 2 \\ 2 & 7 & 1 \\ 2 & 6 & 4 \end{array}\right) $$

Evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{C}=\left(\begin{array}{rrr} -5 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 3 \end{array}\right) $$

Let \(\mathbf{v}_{1}, \mathbf{v}_{2}\), and \(\mathbf{v}_{3}\) be the first, second, and third column vectors, respectively, of the matrix $$ \mathbf{A}=\left(\begin{array}{rrr} 2 & 1 & 7 \\ 1 & 0 & 2 \\ -1 & 5 & 13 \end{array}\right) $$ What can we conclude about \(\operatorname{rank}(\mathbf{A})\) from the observation \(2 \mathbf{v}_{1}+3 \mathbf{v}_{2}-\mathbf{v}_{3}=\mathbf{0} ?\) [Hint: Read the Remarks at the end of this section.]

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

In Problems 1-20, fill in the blanks or answer true/false. If a \(3 \times 3\) matrix \(A\) is diagonalizable, then it possesses three linearly independent eigenvectors._________

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

A nonzero \(n \times n\) matrix \(A\) is said to be nilpotent of index \(m\) if \(m\) is the smallest positive integer for which \(\mathbf{A}^{m}=\mathbf{0}\). Which of the following matrices are nilpotent? If nilpotent, what is its index? (a) \(\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right)\) (b) \(\left(\begin{array}{rr}2 & 2 \\ -2 & -2\end{array}\right)\) (c) \(\left(\begin{array}{lll}0 & 0 & 0 \\ 1 & 0 & 0 \\ 2 & 3 & 0\end{array}\right)\) (d) \(\left(\begin{array}{lll}0 & 0 & 5 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)\) (e) \(\left(\begin{array}{rrrr}0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ -1 & 0 & 0 & 0\end{array}\right)\) (f) \(\left(\begin{array}{llll}0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 2 & 2 & 1 & 0\end{array}\right)\)

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

In Problems 15-18, evaluate the determinant of the given matrix without expanding by cofactors. $$ \mathbf{C}=\left(\begin{array}{rrr} -5 & 0 & 0 \\ 0 & 7 & 0 \\ 0 & 0 & 3 \end{array}\right) $$

In Problems 15-28, evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{lll} 3 & 0 & 2 \\ 2 & 7 & 1 \\ 2 & 6 & 4 \end{array}\right) $$

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

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

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

Evaluate the determinant of the given matrix by cofactor expansion. $$ \left(\begin{array}{rrr} 1 & -1 & -1 \\ 2 & 2 & -2 \\ 1 & 1 & 9 \end{array}\right) $$

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