Chapter 11

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

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.

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

(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 \(A\)

\(1-2\) . 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______=______

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? $$(5,-1), \quad(-1,3), \quad(2,1)$$

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

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

(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 $$A \quad X=B$$ (b) The inverse of \(A\) is \(A^{-1}=\)(c) The solution of the matrix equation is \(X=A^{-1} B\) . $$ X=A^{-1} \quad B $$ $$\left[\begin{array}{l}{x} \\ {y}\end{array}\right]=$$ (d) The solution of the system is \(x=\) _______, \(y=\) ______

(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? \(\begin{array}{llll}{\text { (i) } A B} & {\text { (ii) } B A} & {\text { (iii) } A A} & {\text { (iv) } B B}\end{array}\)

\(1-2\) . 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______=____

Write the augmented matrix of the following system of equations. $$ \begin{aligned} \text { System } & \\\\\left\\{\begin{aligned} x+y-z=& 1 \\\ x+2 z=&-3 \\ 2 y-z=& 3 \end{aligned}\right.\end{aligned} $$ $$ \text { Augmented matrix } $$ Table cannot copy

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

\(3-8=\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{y=x^{2}} \\ {y=x+12}\end{array}\right. $$

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{1}{(x-1)(x+2)} $$

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

Which of the following operations can we perform for a matrix \(A\) of any dimension? \(\begin{array}{llll}{\text { (i) } A+A} & {\text { (ii) } 2 A} & {\text { (iii) } A \cdot A}\end{array}\)

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

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=_____.

Fill in the blanks with appropriate numbers to calculate the determinant. Where there is " " , choose the appropriate sign \((+\text { or }-) .\) (a) \(\left|\begin{array}{rr}{2} & {1} \\ {-3} & {4}\end{array}\right|=\) ______ - ______ = ______ (b) \(\left|\begin{array}{rrr}{1} & {0} & {2} \\ {3} & {2} & {1} \\ {0} & {-3} & {4}\end{array}\right|=\pm\) (___ - ___) \(\pm\) __ (___ - __) \(\pm\) (____ - ____) = ____

\(3-16=\) Graph the inequality. $$ y \geq-2 $$

\(3-8=\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=25 \\ y &=2 x \end{aligned}\right. $$

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

\(3-6\) . State whether the equation or system of equations is linear. $$ x^{2}+y^{2}+z^{2}=4 $$

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

The following is a system of two linear equations in two variables. $$\left\\{\begin{array}{c}{x+y=1} \\ {2 x+2 y=2}\end{array}\right.$$ The graph of the first equation is the same as the graph of the second equation, so the system has _____ _____ solutions. We express these solutions by writing $$\begin{array}{l}{x=t} \\ {y=}\end{array}$$ where \(t\) is any real number. Some of the solutions of this system are (1,_____),(-3,_____), and (5,_____).

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

\(3-16=\) Graph the inequality. $$ y>x $$

\(3-8=\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {x+y=0}\end{array}\right. $$

\(3-12\) . 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. $$ \left[\begin{array}{rr}{2} & {7} \\ {0} & {-1} \\ {5} & {-3}\end{array}\right] $$

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

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

\(3-6\) . State whether the equation or system of equations is linear. $$ \left\\{\begin{aligned} x y-3 y+z &=5 \\ x-y^{2}+5 z &=0 \\ 2 x &+y z=3 \end{aligned}\right. $$

\(5-8\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{aligned} x-y &=1 \\ 4 x+3 y &=18 \end{aligned}\right. $$

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

\(3-8=\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{aligned} x^{2}+y &=9 \\ x-y+3 &=0 \end{aligned}\right. $$

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

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{1}{x^{4}-x^{3}} $$

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

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

\(3-6\) . State whether the equation or system of equations is linear. $$ \left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 y &=2 \\ y+2 z &=4 \end{aligned}\right. $$

\(5-8\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{array}{l}{3 x+y=1} \\ {5 x+2 y=1}\end{array}\right. $$

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

\(3-8=\) Use the substitution method to find all solutions of the system of equations. $$ \left\\{\begin{aligned} x+y^{2} &=0 \\ 2 x+5 y^{2} &=75 \end{aligned}\right. $$

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

\(3-12\) . Write the form of the partial fraction decomposition of the function (as in Example 4 ). Do not determine the numerical values of the coefficients. $$ \frac{x^{2}}{(x-3)\left(x^{2}+4\right)} $$

Perform the matrix operation, or if it is impossible, explain why. $$ \left[\begin{array}{rr}{2} & {6} \\ {-5} & {3}\end{array}\right]+\left[\begin{array}{rr}{-1} & {-3} \\ {6} & {2}\end{array}\right] $$