Chapter 10

To graph an inequality, we first graph the corresponding _______. So to graph \(y \leq x+1,\) we first graph the equation _______. To decide which side of the graph of the equation is the graph of the inequality, we use _______ points. Using \((0,0)\) as such a point, graph the inequality by shading the appropriate region. (GRAPH CAN'T COPY)

For each rational function \(r,\) choose from (i)-(iv) the appropriate form for its partial fraction decomposition. $$r(x)=\frac{4}{x(x-2)^{2}}$$ $$\text { (i) } \frac{A}{x}+\frac{B}{x-2}$$ $$\text { (ii) } \frac{A}{x}+\frac{B}{(x-2)^{2}}$$ $$\text { (iii) } \frac{A}{x}+\frac{B}{x-2}+\frac{C}{(x-2)^{2}}$$ $$\text { (iv) } \frac{A}{x}+\frac{B}{x-2}+\frac{C x+D}{(x-2)^{2}}$$

True or false? \(\operatorname{det}(A)\) is defined only for a square matrix \(A\)

(a) The matrix \(I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]\) is called an _____ matrix. (b) If \(A\) is a \(2 \times 2\) matrix, then \(A \times I=\) _____ and \(I \times A=\) _____. (c) If \(A\) and \(B\) are \(2 \times 2\) matrices with \(A B=I,\) then \(B\) is the _____ of \(A\).

We can add (or subtract) two matrices only if they have the same _________.

If a system of linear equations has infinitely many solutions, then the system is called ______. If a system of linear equations has no solution, then the system is called ______.

The system of equations $$\left\\{\begin{array}{l}2 x+3 y=7 \\\5 x-y=9\end{array}\right.$$ is a system of two equations in the two variables _____ and _____. To determine whether \((5,-1)\) is a solution of this system, we check whether \(x=5\) and \(y=-1\) satisfy each _____ in the system. Which of the following are solutions of this system? $$(5,-1), \quad(-1,3), \quad(2,1)$$

These exercises refer to the following system. $$ \left\\{\begin{aligned} x-y+z=& 2 \\ -x+2 y+z=&-3 \\ 3 x+y-2 z=& 2 \end{aligned}\right. $$ If we add 2 times the first equation to the second equation, the second equation becomes _________ = ________

(a) Write the following system as a matrix equation \(A X=B\) System \(5 x+3 y=4\) \(3 x+2 y=3\) Matrix equation $$A \cdot \quad X=B$$ \(\left[\begin{array}{ll} {\square} & {\square} \\ {\square} & {\square}\end{array}\right]\left[\begin{array}{l}{\square} \\\ {\square}\end{array}\right]=\left[\begin{array}{l}{\square} \\\ {\square}\end{array}\right]\) (b) The inverse of \(A\) is \(A^{-1}=\left[\begin{array}{ll}\square & \square \\\ \square & \square\end{array}\right]\) (c) The solution of the matrix equation is \(X=A^{-1} B\) $$\begin{aligned} &X=A^{-1} \quad B\\\ &\left[\begin{array}{l} x \\ y \end{array}\right]=\left[\begin{array}{ll} \square & \square \\ \square & \square \end{array}\right]\left[\begin{array}{l} \square \\ \square \end{array}\right]=\left[\begin{array}{l} \square \\ \square \end{array}\right] \end{aligned}$$ (d) The solution of the system is \(x=\) _________, \(y=\) _________.

For each rational function \(r,\) choose from (i)-(iv) the appropriate form for its partial fraction decomposition. $$r(x)=\frac{2 x+8}{(x-1)\left(x^{2}+4\right)}$$ (i) \(\frac{A}{x-1}+\frac{B}{x^{2}+4}\) (ii) \(\frac{A}{x-1}+\frac{B x+C}{x^{2}+4}\) (iii) \(\frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{x^{2}+4}\) (iv) \(\frac{A x+B}{x-1}+\frac{C x+D}{x^{2}+4}\)

True or false? \(\operatorname{det}(A)\) is a number, not a matrix.

Write the augmented matrix of the following system of equations. System \(\left\\{\begin{array}{rr}x+y-z= & 1 \\ +2 z= & -3 \\ 2 y-z= & 3\end{array}\right.\) Augmented matrix \(\left[\begin{array}{llll}\text{_} & \text{_}& \text{_} & \text{_} \\ \text{_} & \text{_} & \text{_} & \text{_} \\ \text{_} & \text{_} & \text{_} & \text{_} \end{array}\right]\)

(a) We can multiply two matrices only if the number of ________ in the first matrix is the same as the number of ___________ in the second matrix. (b) If \(A\) is a \(3 \times 3\) matrix and \(B\) is a \(4 \times 3\) matrix, which of the following matrix multiplications are possible? (i) \(A B\) (ii) \(B A\) (iii) \(A A\) (iv) \(B B\)

A system of equations in two variables can be solved by the _____ method, the _____ method, or the _____ method.

These exercises refer to the following system. $$ \left\\{\begin{aligned} x-y+z=& 2 \\ -x+2 y+z=&-3 \\ 3 x+y-2 z=& 2 \end{aligned}\right. $$ To eliminate x from the third equation, we add _______ times the first equation to the third equation. The third equation becomes ________=_______

Graph the inequality. $$x<3$$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{ll} 4 & 1 \\ 7 & 2 \end{array}\right] \quad B=\left[\begin{array}{rr} 2 & -1 \\ -7 & 4 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{array}{l} y=x^{2} \\ y=x+12 \end{array}\right.$$

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{1}{(x-1)(x+2)}$$

The following matrix is the augmented matrix of a system of linear equations in the variables \(x, y,\) and \(z\). (It is given in reduced row-echelon form.) $$\left[\begin{array}{rrrr} 1 & 0 & -1 & 3 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 0 & 0 \end{array}\right]$$ (a) The leading variables are _____. (b) Is the system inconsistent or dependent? _____. (c) The solution of the system is: x=____, y=____, z=____

True or false? If \(\operatorname{det}(A)=0,\) then \(A\) is not invertible.

Which of the following operations can we perform for a matrix \(A\) of any dimension? (i) \(A+A\) (ii) \(2 A\) (iii) \(A \cdot A\)

A system of two linear equations in two variables can have one solution,_____ solution, or _____ _____ solutions.

State whether the equation or system of equations is linear. $$6 x-\sqrt{3} y+\frac{1}{2} z=0$$

Graph the inequality. $$y \geq-2$$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{ll} 2 & -3 \\ 4 & -7 \end{array}\right] \quad B=\left[\begin{array}{ll} \frac{7}{2} & -\frac{3}{2} \\ 2 & -1 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ y &=2 x \end{aligned}\right.$$

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{x}{x^{2}+3 x-4}$$

The augmented matrix of a system of linear equations is given in reduced row- echelon form. Find the solution of the system. $$\begin{aligned} &\text { (a) }\left[\begin{array}{llll} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 3 \end{array}\right]\\\ &x=\text{____}\\\ &y=\text{____}\\\ &z=\text{____} \end{aligned}$$ $$\begin{aligned} &\text { (b) }\left[\begin{array}{llll} 1 & 0 & 1 & 2 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\\\ &\boldsymbol{x}=\\\ &y=\text{____}\\\ &z=\text{____} \end{aligned}$$ $$\begin{aligned} &\text { (c) }\left[\begin{array}{llll} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 3 \end{array}\right]\\\ &x=\text{____}\\\ &y=\text{____}\\\ &z=\text{____} \end{aligned}$$

Fill in the missing entries in the product matrix. $$\left[\begin{array}{rrr} 3 & 1 & 2 \\ -1 & 2 & 0 \\ 1 & 3 & -2 \end{array}\right]\left[\begin{array}{rrr} -1 & 3 & -2 \\ 3 & -2 & -1 \\ 2 & 1 & 0 \end{array}\right]=\left[\begin{array}{rrr} 4 & \mathbb{I} & -7 \\ 7 & -7 & \mathbb{I} \\ -5 & -5 & -5 \end{array}\right]$$

State whether the equation or system of equations is linear. $$x^{2}+y^{2}+z^{2}=4$$

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rrr} 1 & 3 & -1 \\ 1 & 4 & 0 \\ -1 & -3 & 2 \end{array}\right] \quad B=\left[\begin{array}{rrr} 8 & -3 & 4 \\ -2 & 1 & -1 \\ 1 & 0 & 1 \end{array}\right]$$

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{ll} 2 & 0 \\ 0 & 3 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{array}{l} x^{2}+y^{2}=8 \\ x+y=0 \end{array}\right.$$

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{x^{2}-3 x+5}{(x-2)^{2}(x+4)}$$

State the dimension of the matrix. $$\left[\begin{array}{rr} 2 & 7 \\ 0 & -1 \\ 5 & -3 \end{array}\right]$$

Determine whether the matrices \(A\) and \(B\) are equal. $$A=\left[\begin{array}{rrr} 1 & -2 & 0 \\ \frac{1}{2} & 6 & 0 \end{array}\right] \quad B=\left[\begin{array}{rr} 1 & -2 \\ \frac{1}{2} & 6 \end{array}\right]$$

State whether the equation or system of equations is linear. $$\left\\{\begin{aligned} x y-3 y+z &=5 \\ x-y^{2}+5 z &=0 \\ 2 x &+y z=3 \end{aligned}\right.$$

Graph the inequality. $$y

Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\). $$A=\left[\begin{array}{rrr} 3 & 2 & 4 \\ 1 & 1 & -6 \\ 2 & 1 & 12 \end{array}\right] \quad B=\left[\begin{array}{rrr} 9 & -10 & -8 \\ -12 & 14 & 11 \\ -\frac{1}{2} & \frac{1}{2} & \frac{1}{2} \end{array}\right]$$

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 0 & -1 \\ 2 & 0 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{aligned} x^{2}+y &=9 \\ x-y+3 &=0 \end{aligned}\right.$$

Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$\frac{1}{x^{4}-x^{3}}$$

State the dimension of the matrix. $$\left[\begin{array}{rrrr} -1 & 5 & 4 & 0 \\ 0 & 2 & 11 & 3 \end{array}\right]$$

Determine whether the matrices \(A\) and \(B\) are equal. $$A=\left[\begin{array}{cc} \frac{1}{4} & \ln 1 \\ 2 & 3 \end{array}\right] \quad B=\left[\begin{array}{cc} 0.25 & 0 \\ \sqrt{4} & \frac{6}{2} \end{array}\right]$$

State whether the equation or system of equations is linear. $$\left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 \quad &=2 \\ y+2 z &=4 \end{aligned}\right.$$

Graph the inequality. $$y \leq 2 x+2$$

Find the determinant of the matrix, if it exists. $$\left[\begin{array}{rr} 4 & 5 \\ 0 & -1 \end{array}\right]$$

Use the substitution method to find all solutions of the system of equations. $$\left\\{\begin{array}{r} x+y^{2}=0 \\ 2 x+5 y^{2}=75 \end{array}\right.$$

State the dimension of the matrix. $$\left[\begin{array}{l} 12 \\ 35 \end{array}\right]$$