Chapter 8

If \(a, b\), and \(c\) are real numbers and \(c \neq 0\), then \(a c=b c\) implies \(a=b .\) For matrices, \(\mathbf{A C}=\mathbf{B C}, \mathbf{C} \neq \mathbf{0}\), does not necessarily imply \(\mathbf{A}=\mathbf{B}\). Verify this for and $$ \begin{aligned} &\mathbf{A}=\left(\begin{array}{lll} 2 & 1 & 4 \\ 3 & 2 & 1 \\ 1 & 3 & 2 \end{array}\right), \mathbf{B}=\left(\begin{array}{rrr} 5 & 1 & 6 \\ 9 & 2 & -3 \\ -1 & 3 & 7 \end{array}\right) \\ &\mathbf{C}=\left(\begin{array}{lll} 0 & 0 & 0 \\ 2 & 3 & 4 \\ 0 & 0 & 0 \end{array}\right) . \end{aligned} $$

Suppose \(\mathbf{A}\) is a \(5 \times 5\) matrix for which \(\operatorname{det} \mathbf{A}=-7\). What is the value of \(\operatorname{det}(2 \mathbf{A}) ?\)

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

$$ (\mathbf{A}+\mathbf{B})^{2}=\mathbf{A}^{2}+2 \mathbf{A} \mathbf{B}+\mathbf{B}^{2} $$

Suppose \(\mathbf{A}\) and \(\mathbf{B}\) are \(n \times n\) matrices and \(\mathbf{A}\) is nonsingular. Show that if \(\mathbf{A} \mathbf{B}=\mathbf{0}\), then \(\mathbf{B}=\mathbf{0}\).

An \(n \times n\) matrix \(\mathbf{A}\) is said to be skew-symmetric if \(\mathbf{A}^{T}=-\mathbf{A}\). If \(\mathbf{A}\) is a \(5 \times 5\) skew-symmetric matrix, show that \(\operatorname{det} \mathbf{A}=0\).

Solve the system \(2 x_{1}+3 x_{2}-x_{3}=6\) \(x_{1}-2 x_{2}=-3\) \(-2 x_{1}+\quad x_{3}=9\) by writing it as a matrix equation and finding the inverse of the coefficient matrix.

An \(n \times n\) matrix \(A\) is said to be skew-symmetric if \(\mathbf{A}^{T}=-\mathbf{A}\). If \(\mathbf{A}\) is a \(5 \times 5\) skew-symmetric matrix, show that \(\operatorname{det} \mathbf{A}=0\).

$$ (\mathbf{A}+\mathbf{B})(\mathbf{A}-\mathbf{B})=\mathbf{A}^{2}-\mathbf{B}^{2} $$

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

It takes about \(n !\) multiplications to evaluate the determinant of an \(n \times n\) matrix using expansion by cofactors, whereas it takes about \(n^{3} / 3\) arithmetic operations using the row-reduction method. Compare the number of operations for both methods using a \(25 \times 25\) matrix.

Use a CAS to solve the given system. \(x_{1}+2 x_{2}-2 x_{3}=0\) \(2 x_{1}-2 x_{2}+x_{3}=0\) \(3 x_{1}-6 x_{2}+4 x_{3}=0\) \(4 x_{1}+14 x_{2}-13 x_{3}=0\)

Use the inverse of the matrix \(\mathbf{A}\) to solve the system \(\mathbf{A X}=\mathbf{B}\), where $$ A=\left(\begin{array}{lll} 1 & 2 & 3 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{array}\right) $$ and the vector \(\mathbf{B}\) is given by (a) \(\left(\begin{array}{l}1 \\ 1 \\\ 1\end{array}\right)\) (b) \(\left(\begin{array}{r}-2 \\ 1 \\\ 3\end{array}\right)\)

$$ \text { Write }\left(\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right)\left(\begin{array}{l} x_{1} \\ x_{2} \end{array}\right)=\left(\begin{array}{l} b_{1} \\ b_{2} \end{array}\right) \text { without matrices. } $$

If \(\mathbf{A}\) and \(\mathbf{B}\) are nonsingular \(n \times n\) matrices, is \(\mathbf{A}+\mathbf{B}\) necessarily nonsingular?

Use a CAS to solve the given system. \(1.2 x_{1}+3.5 x_{2}-4.4 x_{3}+3.1 x_{4}=1.8\) \(0.2 x_{1}-6.1 x_{2}-2.3 x_{3}+5.4 x_{4}=-0.6\) \(3.3 x_{1}-3.5 x_{2}-2.4 x_{3}-0.1 x_{4}=2.5\) \(5.2 x_{1}+8.5 x_{2}-4.4 x_{3}-2.9 x_{4}=0\)

In Problems 41-46, find the eigenvalues and corresponding eigenvectors of the given matrix. $$ \left(\begin{array}{ll} 1 & 2 \\ 4 & 3 \end{array}\right) $$

In Problems \(39-42\), use a CAS to solve the given system. $$ \begin{aligned} &1.2 x_{1}+3.5 x_{2}-4.4 x_{3}+3.1 x_{4}=1.8 \\ &0.2 x_{1}-6.1 x_{2}-2.3 x_{3}+5.4 x_{4}=-0.6 \\ &3.3 x_{1}-3.5 x_{2}-2.4 x_{3}-0.1 x_{4}=2.5 \\ &5.2 x_{1}+8.5 x_{2}-4.4 x_{3}-2.9 x_{4}=0 \end{aligned} $$

Write the system of equations $$ \begin{aligned} 2 x_{1}+6 x_{2}+x_{3} &=7 \\ x_{1}+2 x_{2}-x_{3} &=-1 \\ 5 x_{1}+7 x_{2}-4 x_{3} &=9 \end{aligned} $$ as a matrix equation \(\mathbf{A} \mathbf{X}=\mathbf{B}\), where \(\mathbf{X}\) and \(\mathbf{B}\) are column vectors.

Consider the \(3 \times 3\) diagonal matrix $$ \mathbf{A}=\left(\begin{array}{rrr} a_{11} & 0 & 0 \\ 0 & a_{22} & 0 \\ 0 & 0 & a_{33} \end{array}\right) . $$ Determine conditions such that \(\mathbf{A}\) is nonsingular. If \(\mathbf{A}\) is nonsingular, find \(\mathbf{A}^{-1}\). Generalize your results to an \(n \times n\) diagonal matrix.

Use a CAS to solve the given system. \(\begin{aligned} x_{1}-x_{2}-x_{3}+2 x_{4}-x_{5} &=5 \\ 6 x_{1}+9 x_{2}-6 x_{3}+17 x_{4}-x_{5} &=40 \\ 2 x_{1}+x_{2}-2 x_{3}+5 x_{4}-x_{5} &=12 \\\ x_{1}+2 x_{2}-x_{3}+3 x_{4} &=7 \\ x_{1}+2 x_{2}+x_{3}+3 x_{4} &=1 \end{aligned}\)

In Problems 41-46, find the eigenvalues and corresponding eigenvectors of the given matrix. $$ \left(\begin{array}{ll} 0 & 0 \\ 4 & 0 \end{array}\right) $$

In Problems \(39-42\), use a CAS to solve the given system. $$ \begin{aligned} x_{1}-x_{2}-x_{3}+2 x_{4}-x_{5} &=5 \\ 6 x_{1}+9 x_{2}-6 x_{3}+17 x_{4}-x_{5} &=40 \\ 2 x_{1}+x_{2}-2 x_{3}+5 x_{4}-x_{5} &=12 \\ x_{1}+2 x_{2}-x_{3}+3 x_{4} &=7 \\ x_{1}+2 x_{2}+x_{3}+3 x_{4} &=1 \end{aligned} $$

Verify that the quadratic form \(a x^{2}+b x y+c y^{2}\) is the same as $$ \left(\begin{array}{ll} x & y \end{array}\left(\begin{array}{rr} a & \frac{1}{2} b \\ \frac{1}{2} b & c \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right) .\right. $$

Use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}+x_{2} &=4 \\ 2 x_{1}-x_{2} &=14 \end{aligned} $$

In Problems 41-46, find the eigenvalues and corresponding eigenvectors of the given matrix. $$ \left(\begin{array}{lll} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & 3 \end{array}\right) $$

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}+x_{2} &=4 \\ 2 x_{1}-x_{2} &=14 \end{aligned} $$

Use an inverse matrix to solve the given system of equations. $$ \begin{gathered} x_{1}-x_{2}=2 \\ 2 x_{1}+4 x_{2}=-5 \end{gathered} $$

In Problems 41-46, find the eigenvalues and corresponding eigenvectors of the given matrix. $$ \left(\begin{array}{rrr} 7 & -2 & 0 \\ -2 & 6 & 2 \\ 0 & 2 & 5 \end{array}\right) $$

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}-x_{2} &=2 \\ 2 x_{1}+4 x_{2} &=-5 \end{aligned} $$

Use an inverse matrix to solve the given system of equations. $$ \begin{aligned} &4 x_{1}-6 x_{2}=6 \\ &2 x_{1}+x_{2}=1 \end{aligned} $$

In Problems 41-46, find the eigenvalues and corresponding eigenvectors of the given matrix. $$ \left(\begin{array}{rrr} -2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{array}\right) $$

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} &4 x_{1}-6 x_{2}=6 \\ &2 x_{1}+x_{2}=1 \end{aligned} $$

Use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}+2 x_{2} &=4 \\ 3 x_{1}+4 x_{2} &=-3 \end{aligned} $$

In Problems 41-46, find the eigenvalues and corresponding eigenvectors of the given matrix. $$ \left(\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 2 & 2 & 1 \end{array}\right) $$

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}+2 x_{2} &=4 \\ 3 x_{1}+4 x_{2} &=-3 \end{aligned} $$

Supply a first column so that the matrix is orthogonal: $$ \left(\begin{array}{c:cc} & -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \\ & 0 & \frac{1}{\sqrt{3}} \\ & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{3}} \end{array}\right) \text {. } $$

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}+x_{3} &=-4 \\ x_{1}+x_{2}+x_{3} &=0 \\ 5 x_{1}-x_{2} &=6 \end{aligned} $$

Use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}-x_{2}+x_{3} &=1 \\ 2 x_{1}+x_{2}+2 x_{3} &=2 \\ 3 x_{1}+2 x_{2}-x_{3} &=-3 \end{aligned} $$

Consider the symmetric matrix \(\mathbf{A}=\left(\begin{array}{rrr}1 & 0 & -2 \\\ 0 & 0 & 0 \\ -2 & 0 & 4\end{array}\right)\). (a) Find matrices \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) that orthogonally diagonalize the matrix \(\mathbf{A}\). (b) Find the diagonal matrix \(D\) by actually carrying out the multiplication \(\mathbf{P}^{-1} \mathbf{A P}\).

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}-x_{2}+x_{3} &=1 \\ 2 x_{1}+x_{2}+2 x_{3} &=2 \\ 3 x_{1}+2 x_{3}-x_{3} &=-3 \end{aligned} $$

Use an inverse matrix to solve the given system of equations. $$ \begin{gathered} x_{1}+2 x_{2}+2 x_{3}=1 \\ x_{1}-2 x_{2}+2 x_{3}=-3 \\ 3 x_{1}-x_{2}+5 x_{3}=7 \end{gathered} $$

$$ \text { Identify the conic section } x^{2}+3 x y+y^{2}=1 \text {. } $$

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}-x_{2}+x_{3} &=1 \\ 2 x_{1}+x_{2}+2 x_{3} &=2 \\ 3 x_{1}+2 x_{3}-x_{3} &=-3 \end{aligned} $$

Use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}-\quad x_{3}-2 &=2 \\ x_{2}+x_{3} &=1 \\ -x_{1}+x_{2}+2 x_{3}+x_{4} &=-5 \\ x_{3}-x_{4} &=3 \end{aligned} $$

In Problems \(43-50\), use an inverse matrix to solve the given system of equations. $$ \begin{aligned} x_{1}-x_{3} &=2 \\ x_{2}+x_{3} &=1 \\ -x_{1}+x_{2}+2 x_{3}+x_{4} &=-5 \\ x_{3}-x_{4} &=3 \end{aligned} $$

Write the system in the form \(\mathbf{A} \mathbf{X}=\mathbf{B}\). Use \(\mathbf{X}=\mathbf{A}^{-1} \mathbf{B}\) to solve the system for each matrix \(\mathbf{B}\). $$ \begin{aligned} &7 x_{1}-2 x_{2}=b_{1} \\ &3 x_{1}-2 x_{2}=b_{2} \\ &\mathbf{B}=\left(\begin{array}{l} 5 \\ 4 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{c} 10 \\ 50 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{r} 0 \\ -20 \end{array}\right) \end{aligned} $$

Write the system in the form \(\mathbf{A} \mathbf{X}=\mathbf{B}\). Use \(\mathbf{X}=\mathbf{A}^{-1} \mathbf{B}\) to solve the system for each matrix \(\mathbf{B}\). $$ \begin{aligned} x_{1}+2 x_{2}+5 x_{3} &=b_{1} \\ 2 x_{1}+3 x_{2}+8 x_{3} &=b_{2} \\ -x_{1}+x_{2}+2 x_{3} &=b_{3} \\ \mathbf{B}=\left(\begin{array}{r} -1 \\ 4 \\ 6 \end{array}\right), \quad \mathbf{B} &=\left(\begin{array}{l} 3 \\ 3 \\ 3 \end{array}\right), & \mathbf{B}=\left(\begin{array}{r} 0 \\ -5 \\ 4 \end{array}\right) \end{aligned} $$

In Problems 51 and 52, write the system in the form \(\mathbf{A X}=\mathbf{B}\). Use \(\mathbf{X}=\mathbf{A}^{-1} \mathbf{B}\) to solve the system for each matrix \(\mathbf{B}\). $$ \begin{aligned} &x_{1}+2 x_{2}+5 x_{3}=b_{1} \\ &2 x_{1}+3 x_{2}+8 x_{3}=b_{2} \\ &-x_{1}+x_{2}+2 x_{3}=b_{3} \\ &\mathbf{B}=\left(\begin{array}{r} -1 \\ 4 \\ 6 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{l} 3 \\ 3 \\ 3 \end{array}\right), \quad \mathbf{B}=\left(\begin{array}{r} 0 \\ -5 \\ 4 \end{array}\right) \end{aligned} $$

Without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution. $$ \begin{array}{r} x_{1}+2 x_{2}-x_{3}=0 \\ 4 x_{1}-x_{2}+x_{3}=0 \\ 5 x_{1}+x_{2}-2 x_{3}=0 \end{array} $$