Chapter 9

Reduce the system of linear equations to upper triangular form and solve. $$ \begin{array}{r} x-2 y=2 \\ 4 y-2 x=-4 \end{array} $$

Find the dot product of \(\mathbf{x}=[-1,2]^{\prime}\) and \(\mathbf{y}=[-3,-4]^{\prime}\)

. Suppose a vector \(\left[\begin{array}{l}x_{1} \\ x_{2}\end{array}\right]\) has length 4 and is \(70^{\circ}\) counterclockwise from the negative \(x_{2}\) -axis. Find \(x_{1}\) and \(x_{2}\).

Suppose that \(A\) and \(B\) are \(m \times n\) matrices. Show that $$ (A+B)^{\prime}=A^{\prime}+B^{\prime} $$

Zach wants to buy fish and plants for his aquarium. Each fish costs \(\$ 2.30 ;\) each plant costs \(\$ 1.70 .\) He buys a total of 11 items and spends a total of \(\$ 21.70 .\) Set up a system of linear equations that will allow you to determine how many fish and how many plants Zach bought, and solve the system.

Find the dot product of \(\mathbf{x}=[0,-1,3]^{\prime}\) and \(\mathbf{y}=[-3,1,1]^{\prime}\).

In Problems 17-22, find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] $$

Suppose that \(A\) is an \(m \times n\) matrix. Show that $$ \left(A^{\prime}\right)^{\prime}=A $$

Eind the dot product of \(\mathbf{x}=[2,-3,1]^{\prime}\) and \(\mathbf{v}=[3,1,-2]^{\prime}\)

In Problems , find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{r} -1 \\ 0 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{l} 3 \\ 2 \end{array}\right] $$

Suppose that \(A\) is an \(m \times n\) matrix and \(k\) is a real number. Show that $$ (k A)^{\prime}=k A^{\prime} $$

Show that if $$ a_{11} a_{22}-a_{21} a_{12} \neq 0 $$ then the system $$ \begin{array}{l} a_{11} x_{1}+a_{12} x_{2}=0 \\ a_{21} x_{1}+a_{22} x_{2}=0 \end{array} $$ has exactly one solution, namely, \(x_{1}=0\) and \(x_{2}=0\).

Use the dot product to compute the length of \([0,-1,2]^{\prime}\).

In Problems , find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{r} 0 \\ -2 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{r} 1 \\ -1 \end{array}\right] $$

Suppose that \(A\) is an \(m \times k\) matrix and \(B\) is a \(k \times n\) matrix. Show that $$ (A B)^{\prime}=B^{\prime} A^{\prime} $$

Solve each system of linear equations. $$ \begin{aligned} 2 x-3 y+z &=-1 \\ x+y-2 z &=-3 \\ 3 x-2 y+z &=2 \end{aligned} $$

Use the dot product to compute the length of \([-1,4,3]\).

In Problems , find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{l} -1 \\ -1 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{l} 1 \\ 2 \end{array}\right] $$

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \begin{array}{l} \text { Compute the following: }\\\ A B \quad \text { (b) } \underline{B A} \end{array} $$

Solve each system of linear equations. $$ \begin{aligned} 5 x-y+2 z &=6 \\ x+2 y-z &=-1 \\ 3 x+2 y-2 z &=1 \end{aligned} $$

Use the dot product to compute the length of \([1,2,3,4]^{\prime}\).

In Problems , find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and \(\mathbf{v}\) $$ \mathbf{x}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{r} -1 \\ 0 \end{array}\right] $$

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \text { Compute } A B C $$

Solve each system of linear equations. $$ \begin{array}{rr} x+4 y-3 z= & -13 \\ 2 x-3 y+5 z= & 18 \\ 3 x+y-2 z= & 1 \end{array} $$

In Problems , find \(\mathbf{x}+\mathbf{y}\) for the given vectors \(\mathbf{x}\) and \(\mathbf{y}\). Represent \(\mathbf{x}, \mathbf{y}\), and \(\mathbf{x}+\mathbf{y}\) in the plane, and explain graphically how vou add \(\mathbf{x}\) and $\mathbf{v}$$$ \mathbf{x}=\left[\begin{array}{l} -3 \\ -1 \end{array}\right] \text { and } \mathbf{y}=\left[\begin{array}{r} -2 \\ 3 \end{array}\right] $$

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \text { Show that } A C \neq C A \text { . } $$

Solve each system of linear equations. $$ \begin{array}{r} -2 x+4 y-z=-1 \\ x+7 y+2 z=-4 \\ 3 x-2 y+3 z=-3 \end{array} $$

Find the angle between \(\mathbf{x}=[1,2]^{\prime}\) and \(\mathbf{y}=[3,-1]^{\prime}\).

In Problems \(23-28\), compute a \(\mathbf{x}\) for the given vector \(\mathbf{x}\) and scalar a. Represent \(\mathbf{x}\) and a \(\mathbf{x}\) in the plane, and explain graphically how you obtain \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} -2 \\ 1 \end{array}\right] \text { and } a=2 $$

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \text { Show that }(A B) C=A(B C) \text { . } $$

Solve each system of linear equations. $$ \begin{array}{l} 2 x-y+3 z=3 \\ 2 x+y+4 z=4 \\ 2 x-3 y+2 z=2 \end{array} $$

Find the angle between \(\mathbf{x}=[-1,2]^{\prime}\) and \(\mathbf{y}=[-2,-4]^{\prime}\).

In Problems , compute a \(\mathbf{x}\) for the given vector \(\mathbf{x}\) and scalar a. Represent \(\mathbf{x}\) and a \(\mathbf{x}\) in the plane, and explain graphically how you obtain \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} 3 \\ -1 \end{array}\right] \text { and } a=-1 $$

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \text { Show that }(A+B) C=A C+B C \text { . } $$

Find the augmented matrix and use it to solve the system of linear equations. $$ \begin{aligned} -x-2 y+3 z &=-9 \\ 2 x+y-z &=5 \\ 4 x-3 y+5 z &=-9 \end{aligned} $$

Find the angle between \(\mathbf{x}=[0,-1,3]^{\prime}\) and \(\mathbf{y}=[-3,1,1]^{\prime}\).

In Problems , compute a \(\mathbf{x}\) for the given vector \(\mathbf{x}\) and scalar a. Represent \(\mathbf{x}\) and a \(\mathbf{x}\) in the plane, and explain graphically how you obtain \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} 0 \\ -2 \end{array}\right] \text { and } a=0.5 $$

$$ A=\left[\begin{array}{rr} -1 & 0 \\ 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 2 & 3 \\ -1 & 1 \end{array}\right], \quad C=\left[\begin{array}{rr} 1 & 2 \\ 0 & -1 \end{array}\right] $$ $$ \text { Show that } A(B+C)=A B+A C $$

Find the augmented matrix and use it to solve the system of linear equations. $$ \begin{array}{l} 3 x-2 y+z=4 \\ 4 x+y-2 z=-12 \\ 2 x-3 y+z=7 \end{array} $$

Find the angle between \(\mathbf{x}=[1,-3,2]^{\prime}\) and \(\mathbf{y}=[3,1,-4]^{\prime}\).

In Problems , compute a \(\mathbf{x}\) for the given vector \(\mathbf{x}\) and scalar a. Represent \(\mathbf{x}\) and a \(\mathbf{x}\) in the plane, and explain graphically how you obtain \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} 3 \\ -9 \end{array}\right] \text { and } a=-1 / 3 $$

Suppose that \(A\) is a \(3 \times 4\) matrix and \(B\) is a \(4 \times 2\) matrix. What is the size of the product \(A B ?\)

Find the augmented matrix and use it to solve the system of linear equations. $$ \begin{array}{l} y+x=3 \\ z-y=-1 \\ x+z=2 \end{array} $$

Let \(\mathbf{x}=[1,-1]^{\prime}\). Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

In Problems , compute a \(\mathbf{x}\) for the given vector \(\mathbf{x}\) and scalar a. Represent \(\mathbf{x}\) and a \(\mathbf{x}\) in the plane, and explain graphically how you obtain \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{r} -4 \\ 1 \end{array}\right] \text { and } a=1 / 4 $$

Suppose \(A\) is a \(3 \times 4\) matrix and \(B\) is an \(m \times n\) matrix. What are values of \(m\) and \(n\) such that the following products are defined? (a) \(A B\) (b) \(B A\)

Find the augmented matrix and use it to solve the system of linear equations. $$ \begin{aligned} 2 x-z &=1 \\ y+3 z &=x-1 \\ x+z &=y-3 \end{aligned} $$

Let \(\mathbf{x}=[-2,1]^{\prime}\). Find \(\mathbf{y}\) so that \(\mathbf{x}\) and \(\mathbf{y}\) are perpendicular.

In Problems , compute a \(\mathbf{x}\) for the given vector \(\mathbf{x}\) and scalar a. Represent \(\mathbf{x}\) and a \(\mathbf{x}\) in the plane, and explain graphically how you obtain \(a \mathbf{x}\). $$ \mathbf{x}=\left[\begin{array}{l} 0.5 \\ 0.25 \end{array}\right] \text { and } a=5 $$

Suppose that \(A\) is a \(4 \times 3\) matrix, \(B\) is a \(1 \times 3\) matrix, \(C\) is a \(3 \times 1\) matrix, and \(D\) is a \(4 \times 3\) matrix. Which of the matrix multiplications that follow are defined? Whenever it is defined, state the size of the resulting matrix.