Chapter 10

Choose from (i)-(iv) the appropriate form for its partial fraction decomposition. $$r(x)=\frac{4}{x(x-2)^{2}}$$ (i) \(\frac{A}{x}+\frac{B}{x-2}\) (ii) \(\frac{A}{x}+\frac{B}{(x-2)^{2}}\) (iii) \(\frac{A}{x}+\frac{B}{x-2}+\frac{C}{(x-2)^{2}}\) (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 \(A\)

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 _________.

These exercises refer to the following system: $$\left\\{\begin{array}{rr}x-y+z= & 2 \\\\-x+2 y+z= & -3 \\\3 x+y-2 z= & 2\end{array}\right.$$ 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 __________.

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

If the point \((2,3)\) is a solution of an inequality in \(x\) and \(y\) then the inequality is satisfied when we replace \(x\) by ____ and \(y\) by ______ Is the point \((2,3)\) a solution of the inequality \(4 x-2 y \geq 1 ?\)

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{aligned} x+y-z &=1 \\ x &+2 z=-3 \\ 2 y-z &=3 \end{aligned}\right. Augmented matrix (EQUATION CANNOT COPY)

(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) \(B B\)

These exercises refer to the following system: $$\left\\{\begin{array}{rr}x-y+z= & 2 \\\\-x+2 y+z= & -3 \\\3 x+y-2 z= & 2\end{array}\right.$$ To eliminate \(x\) from the third equation, we add ____________ times the first equation to the third equation. The third equation becomes ___________ \(=\) ______.

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

If the point \((2,3)\) is a solution of a system of inequalities in \(x\) and \(y,\) then each inequality is satisfied when we replace \(x\) by ______ and \(y\) by _______. Is the point \((2,3)\) a solution of the following system? $$\left\\{\begin{array}{l} 2 x+4 y \leq 17 \\ 6 x+5 y \leq 29 \end{array}\right.$$

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

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

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) \(A \cdot A\)

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: $$x=$$__________ , $$y=$$ __________ , $$=$$ __________

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

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

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

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] \quad \text { (b) }\left[\begin{array}{llll} 1 & 0 & 1 & 2 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] \quad \text { (c) }\left[\begin{array}{llll} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 3 \end{array}\right] $$x=$$_____ $$y=$$_____ $$z=$$_____ $$x=$$_____ $$y=$$_____ $$z=$$_____ $$x=$$_____ $$y=$$_____ $$z=$$_____

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

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

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 $$\begin{array}{l} x=t \\ y=\text{_____} \end{array}$$ where \(t\) is any real number. Some of the solutions of this system are (1,___),(-3,___), and (5,___).

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

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

An inequality and several points are given. For each point determine whether it is a solution of the inequality. $$x-5 y>3 ; \quad(-1,-2),(1,-2),(1,2),(8,1)$$

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

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

Verifying the Inverse of a Matrix 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. $$\left[\begin{array}{ll} 2 & 0 \\ 0 & 3 \end{array}\right]$$

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. \(\left\\{\begin{aligned} x y-3 y+z &=5 \\ x-y^{2}+5 z &=0 \\ 2 x &+y z=3 \end{aligned}\right.\)

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

An inequality and several points are given. For each point determine whether it is a solution of the inequality. $$3 x+2 y \leq 2 ; \quad(-2,1),(1,3),(1,-3),(0,1)$$

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

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

Verifying the Inverse of a Matrix 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. $$\left[\begin{array}{rr} 0 & -1 \\ 2 & 0 \end{array}\right]$$

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. \(\left\\{\begin{aligned} x-2 y+3 z &=10 \\ 2 x+5 y &=2 \\ y+2 z &=4 \end{aligned}\right.\)

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

A system of inequalities and several points are given. Determine which points are solutions of the system. $$\left\\{\begin{array}{l} 3 x-2 y \leq 5 \\ 2 x+y \geq 3 \end{array} ; \quad(0,0),(1,2),(1,1),(3,1)\right.$$

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

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