Chapter 8

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

Suppose \(\lambda\) is an eigenvalue with corresponding eigenvector \(\mathbf{K}\) of an \(n \times n\) matrix \(\mathbf{A}\). (a) If \(\mathbf{A}^{2}=\mathbf{A} \mathbf{A}\), then show that \(\mathbf{A}^{2} \mathbf{K}=\lambda^{2} \mathbf{K}\). Explain the meaning of the last equation. (b) Verify the result obtained in part (a) for the matrix \(\mathbf{A}=\left(\begin{array}{ll}2 & 3 \\ 5 & 4\end{array}\right)\)

Determine the size of the matrix \(\mathbf{A}\) such that the given product is defined. $$ \left(\begin{array}{rrr} 2 & 1 & 3 \\ 3 & 9 & 6 \\ 7 & 0 & -1 \end{array}\right) \mathbf{A}\left(\begin{array}{ll} 0 & 1 \\ 7 & 4 \end{array}\right) $$

If \(\mathbf{A}\) is nonsingular, then \(\left(\mathbf{A}^{T}\right)^{-1}=\left(\mathbf{A}^{-1}\right)^{T}\). Verify this for \(\mathbf{A}=\left(\begin{array}{rr}1 & 4 \\ 2 & 10\end{array}\right) .\)

Find the values of \(\lambda\) that satisfy the given equation. $$ \left|\begin{array}{ccr} 1-\lambda & 0 & -1 \\ 1 & 2-\lambda & 1 \\ 3 & 3 & -\lambda \end{array}\right|=0 $$

In Problems 29 and 30, evaluate the determinant of the given matrix by inspection. $$ \left(\begin{array}{rrrr} -3 & 0 & 0 & 0 \\ 4 & 6 & 0 & 0 \\ 1 & 3 & 9 & 0 \\ 6 & 4 & 2 & 1 \end{array}\right) $$

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

In Problems 29 and 30, find the values of \(\lambda\) that satisfy the given equation. $$ \left|\begin{array}{ccr} 1-\lambda & 0 & -1 \\ 1 & 2-\lambda & 1 \\ 3 & 3 & -\lambda \end{array}\right|=0 $$

$$, Suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\) Verify the given property by computing the left and right members of the given equality. $$ \left(\mathbf{A}^{T}\right)^{T}=\mathbf{A} $$

Find a value of \(x\) such that the matrix \(\mathbf{A}=\left(\begin{array}{ll}4 & -3 \\ x & -4\end{array}\right)\) is its own inverse.

An elementary matrix \(\mathbf{E}\) is one obtained by performing a single row operation on the identity matrix I. Verify that the given matrix is an elementary matrix. \(\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)\)

In Problems 31 and 32, without solving, state whether the given homogeneous system has only the trivial solution or has infinitely many solutions. $$ \begin{array}{r} x_{1}-x_{2}+x_{3}=0 \\ 5 x_{1}+x_{2}-x_{3}=0 \\ x_{1}+2 x_{2}+x_{3}=0 \end{array} $$

An \(n \times n\) matrix \(A\) is said to be a stochastic matrix if all its entries are nonnegative and the sum of the entries in each row (or the sum of the entries in each column) add up to \(1 .\) Stochastic matrices are important in probability theory. (a) Verify that $$ \mathbf{A}=\left(\begin{array}{ll} p & 1-p \\ q & 1-q \end{array}\right), \quad 0 \leq p \leq 1,0 \leq q \leq 1 $$ and $$ \mathbf{A}=\left(\begin{array}{ccc} \frac{1}{2} & \frac{1}{4} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{6} & \frac{1}{3} & \frac{1}{2} \end{array}\right) $$ are stochastic matrices. (b) Use a CAS or linear algebra software to find the eigenvalues and eigenvectors of the the \(3 \times 3\) matrix \(A\) in part (a). Make up at least six more stochastic matrices of various sizes, \(2 \times 2,3 \times 3,4 \times 4\), and \(5 \times 5\). Find the eigenvalues and eigenvectors of each matrix. If you discern a pattern, form a conjecture and then try to prove it. (c) For the \(3 \times 3\) matrix \(A\) in part (a), use the software to find \(\mathbf{A}^{2}, \mathbf{A}^{3}, \mathbf{A}^{4}, \ldots\) Repeat for the matrices that you constructed in part (b). If you discern a pattern, form a conjecture and then try to prove it.

In Problems 31-34, suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\). Verify the given property by computing the left and right members of the given equality. $$ \left(\mathbf{A}^{T}\right)^{T}=\mathbf{A} $$

$$, Suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\) Verify the given property by computing the left and right members of the given equality. $$ (\mathbf{A B})^{T}=\mathbf{B}^{T} \mathbf{A}^{T} $$

Find the inverse of \(\mathbf{A}=\left(\begin{array}{cc}\sin \theta & \cos \theta \\ -\cos \theta & \sin \theta\end{array}\right)\).

An elementary matrix \(\mathbf{E}\) is one obtained by performing a single row operation on the identity matrix I. Verify that the given matrix is an elementary matrix. \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c\end{array}\right)\)

In Problems 31 and 32, without solving, state whether the given homogeneous system has only the trivial solution or has infinitely many solutions. $$ \begin{array}{r} x_{1}-x_{2}-x_{3}=0 \\ 5 x_{1}+x_{2}-x_{3}=0 \\ x_{1}+2 x_{2}+x_{3}=0 \end{array} $$

$$ \text { Find the inverse of } \mathbf{A}=\left(\begin{array}{cc} \sin \theta & \cos \theta \\ -\cos \theta & \sin \theta \end{array}\right) $$

In Problems 31-34, suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\). Verify the given property by computing the left and right members of the given equality. $$ (\mathbf{A}+\mathbf{B})^{T}=\mathbf{A}^{T}+\mathbf{B}^{T} $$

Use the given LU-factorization $$ \mathbf{A}=\left(\begin{array}{lll} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 2 & 3 \end{array}\right)=\left(\begin{array}{lll} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 1 & 1 \end{array}\right)\left(\begin{array}{lll} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{array}\right) $$ to solve the linear system \(\mathbf{A X}=\mathbf{B}_{i}\) for the given column matrix \(\mathbf{B}_{i}, i=1,2,3,4\) $$ \mathbf{B}_{3}=\left(\begin{array}{r} \frac{1}{2} \\ \frac{3}{4} \\ -\frac{1}{2} \end{array}\right) $$

$$, Suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\) Verify the given property by computing the left and right members of the given equality. $$ (\mathbf{A B})^{T}=\mathbf{B}^{T} \mathbf{A}^{T} $$

An elementary matrix \(\mathbf{E}\) is one obtained by performing a single row operation on the identity matrix I. Verify that the given matrix is an elementary matrix. \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & c & 1\end{array}\right)\)

In Problems 33 and 34, use Gauss-Jordan elimination to balance the given chemical equation. $$ \mathrm{I}_{2}+\mathrm{HNO}_{3} \rightarrow \mathrm{HIO}_{3}+\mathrm{NO}_{2}+\mathrm{H}_{2} \mathrm{O} $$

In Problems 31-34, suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\). Verify the given property by computing the left and right members of the given equality. $$ (\mathbf{A B})^{T}=\mathbf{B}^{T} \mathbf{A}^{T} $$

$$, Suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\) Verify the given property by computing the left and right members of the given equality. $$ (6 \mathbf{A})^{T}=6 \mathbf{A}^{T} $$

Evaluate \(\left|\begin{array}{cccc}1 & 1 & 1 & 1 \\ a & b & c & d \\ a^{2} & b^{2} & c^{2} & d^{2} \\ a^{3} & b^{3} & c^{3} & d^{3}\end{array}\right| .\)

An elementary matrix \(\mathbf{E}\) is one obtained by performing a single row operation on the identity matrix I. Verify that the given matrix is an elementary matrix. \(\left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1\end{array}\right)\)

In Problems 33 and 34, use Gauss-Jordan elimination to balance the given chemical equation. $$ \mathrm{Ca}+\mathrm{H}_{3} \mathrm{PO}_{4} \rightarrow \mathrm{Ca}_{3} \mathrm{P}_{2} \mathrm{O}_{8}+\mathrm{H}_{2} $$

In Problems 31-34, suppose \(\mathbf{A}=\left(\begin{array}{rr}2 & 4 \\ -3 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}4 & 10 \\ 2 & 5\end{array}\right)\). Verify the given property by computing the left and right members of the given equality. $$ (6 A)^{T}=6 A^{T} $$

Find a \(2 \times 2\) matrix A that has eigenvalues \(\lambda_{1}=2\) and \(\lambda_{2}=3\) and corresponding eigenvectors $$ \mathbf{K}_{1}=\left(\begin{array}{l} 1 \\ 2 \end{array}\right) \quad \text { and } \quad \mathbf{K}_{2}=\left(\begin{array}{l} 1 \\ 1 \end{array}\right) $$

$$ \begin{aligned} &\text { Suppose } \mathbf{A}=\left(\begin{array}{ll} 2 & 1 \\ 6 & 3 \\ 2 & 5 \end{array}\right) \text { . Verify that the matrix } \mathbf{B}=\mathbf{A} \mathbf{A}^{T} \text { is }\\\ &\text { symmetric. } \end{aligned} $$

Verify Theorem \(8.5 .9\) by evaluating \(a_{21} C_{11}+a_{22} C_{12}+a_{23} C_{13}\) and \(a_{13} C_{12}+a_{23} C_{22}+a_{33} C_{32}\) for the given matrix. $$ \mathbf{A}=\left(\begin{array}{rrr} 1 & 1 & 2 \\ -1 & 2 & 1 \\ 4 & -2 & 1 \end{array}\right) $$

If a matrix \(\mathbf{A}\) is premultiplied by an elementary matrix \(\mathbf{E}\), the product EA will be that matrix obtained from A by performing the elementary row operation symbolized by \(\mathbf{E}\). Compute the given product for an arbitrary \(3 \times 3\) \(\operatorname{matrix} \mathbf{A}\). \(\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right) \mathbf{A}\)

In Problems 35 and 36 , solve the given system of equations by Cramer's rule. $$ \begin{aligned} x_{1}+2 x_{2}-3 x_{3} &=-2 \\ 2 x_{1}-4 x_{2}+3 x_{3} &=0 \\ 4 x_{2}+6 x_{3} &=5 \end{aligned} $$

If \(\mathbf{A}\) and \(\mathbf{B}\) are nonsingular \(n \times n\) matrices, use Theorem 8.6.3 to show that \(\mathbf{A B}\) is nonsingular.

Find a \(3 \times 3\) symmetric matrix that has eigenvalues \(\lambda_{1}=1\), \(\lambda_{2}=3\), and \(\lambda_{3}=5\) and corresponding eigenvectors $$ \mathbf{K}_{1}=\left(\begin{array}{r} 1 \\ -1 \\ 1 \end{array}\right), \mathbf{K}_{2}=\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right), \text { and } \mathbf{K}_{3}=\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right) $$

$$ \text { Show that if } \mathbf{A} \text { is an } m \times n \text { matrix, then } \mathbf{A A}^{T} \text { is symmetric. } $$

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

If a matrix \(\mathbf{A}\) is premultiplied by an elementary matrix \(\mathbf{E}\), the product EA will be that matrix obtained from A by performing the elementary row operation symbolized by \(\mathbf{E}\). Compute the given product for an arbitrary \(3 \times 3\) \(\operatorname{matrix} \mathbf{A}\). \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & c\end{array}\right) \mathbf{A}\)

In Problems 35 and 36 , solve the given system of equations by Cramer's rule. $$ \begin{aligned} x_{1}+& x_{3}=4 \\ 2 x_{1}+3 x_{2}+4 x_{3} &=5 \\ x_{1}+4 x_{2}+5 x_{3} &=0 \end{aligned} $$

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

If \(\mathbf{A}\) is an \(n \times n\) diagonalizablematrix, then \(\mathbf{D}=\mathbf{P}^{-1} \mathbf{A} \mathbf{P}\), where \(\mathbf{D}\) is a diagonal matrix. Show that if \(m\) is a positive integer, then \(\mathbf{A}^{m}=\mathbf{P D}^{m} \mathbf{P}^{-1}\).

In matrix theory, many of the familiar properties of the real number system are not valid. If \(a\) and \(b\) are real numbers, then \(a b=0\) implies that \(a=0\) or \(b=0 .\) Find two matrices such that \(\mathbf{A B}=\mathbf{0}\) but \(\mathbf{A} \neq \mathbf{0}\) and \(\mathbf{B} \neq \mathbf{0}\)

Show that if \(\mathbf{A}\) is a nonsingular matrix, then \(\operatorname{det} \mathbf{A}^{-1}=1 /\) det \(\mathbf{A}\).

Let \(\mathbf{A}=\left(\begin{array}{rr}3 & -4 \\ 1 & 2\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rr}7 & 4 \\ -1 & -5\end{array}\right)\). Verify that \(\operatorname{det}(\mathbf{A}+\mathbf{B}) \neq \operatorname{det} \mathbf{A}+\operatorname{det} \mathbf{B}\).

If a matrix \(\mathbf{A}\) is premultiplied by an elementary matrix \(\mathbf{E}\), the product EA will be that matrix obtained from A by performing the elementary row operation symbolized by \(\mathbf{E}\). Compute the given product for an arbitrary \(3 \times 3\) \(\operatorname{matrix} \mathbf{A}\). \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & c & 1\end{array}\right) \mathbf{A}\)

Use Cramer's rule to solve the system $$ \begin{aligned} &X=x \cos \theta+y \sin \theta \\ &Y=-x \sin \theta+y \cos \theta \end{aligned} $$ for \(x\) and \(y\).

$$ \text { Show that if } \mathbf{A} \text { is a nonsingular matrix, then } \operatorname{det} \mathbf{A}^{-1}=1 / \operatorname{det} \mathbf{A} \text {. } $$

The \(m\) th power of a diagonal matrix is $$ \begin{gathered} \mathbf{D}=\left(\begin{array}{cccc} a_{11} & 0 & \cdots & 0 \\ 0 & a_{22} & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & a_{n n} \end{array}\right) \\ \mathbf{D}^{m}=\left(\begin{array}{cccc} a_{11}^{m} & 0 & \cdots & 0 \\ 0 & a_{22}^{m} & \cdots & 0 \\ \vdots & & & \vdots \\ 0 & 0 & \cdots & a_{n n}^{m} \end{array}\right) . \end{gathered} $$ Use this result to compute $$ \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 5 \end{array}\right)^{4} $$