Chapter 6

Graph the solution set to the inequality. $$ x \geq y $$

Determine if the matrix \(A\) is invertible by cal. culating det \(A\) $$ A=\left[\begin{array}{ll} 4 & 3 \\ 5 & 4 \end{array}\right] $$

Determine if \(B\) is the inverse matrix of \(A\) by calculating \(A B\) and \(B A\) $$ A=\left[\begin{array}{ll} 4 & 3 \\ 5 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 4 & -3 \\ -5 & 4 \end{array}\right] $$

State the dimension of each matrix. (a) \(\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\) (b) \(\left[\begin{array}{lll}a & b & c \\ d & e & b\end{array}\right]\) (c) \(\left[\begin{array}{rr}3 & 0 \\ 1 & -4\end{array}\right]\)

Evaluate the function for the indicated inputs and interpret the result. \(A(5,8),\) where \(A(b, h)=\frac{1}{2} b h(A\) computes the area of a triangle with base \(b\) and height \(h\) )

Determine each of the following for the given matrix \(A,\) if possible. (a) \(a_{12}, a_{21},\) and \(a_{32} (b) \)a_{11} a_{22}+3 a_{23}$ $$\left[\begin{array}{rrrr}1 & 2 & 3 & 4 \\\5 & 6 & 7 & 8 \\\9 & 10 & 11 & 12\end{array}\right]$$

Graph the solution set to the inequality. $$ y>-3 $$

Determine if the matrix \(A\) is invertible by cal. culating det \(A\) $$ A=\left[\begin{array}{rr} 1 & -3 \\ 2 & 6 \end{array}\right] $$

Determine if \(B\) is the inverse matrix of \(A\) by calculating \(A B\) and \(B A\) $$ A=\left[\begin{array}{ll} -1 & 2 \\ -3 & 8 \end{array}\right], \quad B=\left[\begin{array}{ll} -4 & 1 \\ -2 & 0.5 \end{array}\right] $$

State the dimension of each matrix. \(\begin{array}{ll}\text { (a) }\left[\begin{array}{llll}-1 & 1\end{array}\right] & \text { (b) }\left[\begin{array}{rr}1 & -1 \\ 7 & 5 \\\ -4 & 0\end{array}\right]\end{array}\) (c) \(\left[\begin{array}{rrrr}1 & 3 & 8 & -3 \\ 1 & -1 & 1 & -2 \\ 4 & 5 & 0 & -1\end{array}\right]\)

Evaluate the function for the indicated inputs and interpret the result. \(A(20,35),\) where \(A(w, l)=w l(A\) computes the area of a rectangle with width \(w\) and length \(L\) )

Does the ordered triple \((1,2,3)\) satisfy the equation \(3 x+2 y+z=10 ?\)

Determine each of the following for the given matrix \(A,\) if possible. (a) \(a_{12}, a_{21},\) and \(a_{32} (b) \)a_{11} a_{22}+3 a_{23}$ $$\left[\begin{array}{rrr}1 & -1 & 4 \\\3 & -2 & 5 \\\7 & 0 & -6\end{array}\right]$$

Graph the solution set to the inequality. $$ x<1 $$

Determine if the matrix \(A\) is invertible by cal. culating det \(A\) $$ A=\left[\begin{array}{rr} -4 & 6 \\ -8 & 12 \end{array}\right] $$

Determine if \(B\) is the inverse matrix of \(A\) by calculating \(A B\) and \(B A\) $$ A=\left[\begin{array}{rrr} 1 & -1 & 2 \\ 0 & 1 & -1 \\ 1 & 0 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 2 & -1 \\ -1 & 0 & 1 \\ -1 & -1 & 1 \end{array}\right] $$

Represent the linear system by an augmented matrix, and state the dimension of the matrix. $$ \begin{array}{l} 5 x-2 y=3 \\ -x+3 y=-1 \end{array} $$

Evaluate the expression for the given \(f(x, y)\). $$ f(2,-3) \text { if } f(x, y)=x^{2}+y^{2} $$

To solve a system of linear equations in two variables, how many equations do you usually need?

Graph the solution set to the inequality. $$ y>2 x $$

Determine if the matrix \(A\) is invertible by cal. culating det \(A\) $$ A=\left[\begin{array}{rr} 10 & -20 \\ -5 & 10 \end{array}\right] $$

Determine if \(B\) is the inverse matrix of \(A\) by calculating \(A B\) and \(B A\) $$ A=\left[\begin{array}{rrr} 2 & 1 & 1 \\ -1 & 0 & -1 \\ 0 & 2 & -1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & -1 \\ -1 & -2 & 1 \\ -2 & -4 & 1 \end{array}\right] $$

Represent the linear system by an augmented matrix, and state the dimension of the matrix. $$ \begin{array}{l} 3 x+y=4 \\ -x+4 y=5 \end{array} $$

Evaluate the expression for the given \(f(x, y)\). $$ f(-1,3) \text { if } f(x, y)=2 x^{2}-y^{2} $$

To solve a system of linear equations in three variables, how many equations do you usually need?

If possible, find values for \(x\) and \(y\) so that the matrices \(A\) and \(B\) are equal. $$A=\left[\begin{array}{rr}x & 2 \\\\-2 & 1\end{array}\right]$$ $$B=\left[\begin{array}{rr}1 & 2 \\\\-2 & y\end{array}\right]$$

Graph the solution set to the inequality. $$ x+y \leq 2 $$

Find the specified minor and cofactor for \(A\). $$ M_{12} \text { and } A_{12} \text { if } A=\left[\begin{array}{rrr} 1 & -1 & 3 \\ 2 & 3 & -2 \\ 0 & 1 & 5 \end{array}\right] $$

Determine if \(B\) is the inverse matrix of \(A\) by calculating \(A B\) and \(B A\) $$ A=\left[\begin{array}{rrr} 2 & 1 & -1 \\ 3 & 0 & 2 \\ -1 & 0 & 1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 0 & 1 & -2 \\ 1 & -3 & 7 \\ 0 & -1 & 3 \end{array}\right] $$

Represent the linear system by an augmented matrix, and state the dimension of the matrix. $$ \begin{aligned} -3 x+2 y+z &=-4 \\ 5 x &=9 \\ x-3 y-6 z &=-9 \end{aligned} $$

Evaluate the expression for the given \(f(x, y)\). $$ f(-2,3) \text { if } f(x, y)=3 x-4 y $$

Determine whether each ordered triple is a solution to the system of linear equations. $$ \begin{array}{c} (0,2,-2),(-1,3,-2) \\ x+y-z=4 \\ -x+y+z=2 \\ x+y+z=0 \end{array} $$

If possible, find values for \(x\) and \(y\) so that the matrices \(A\) and \(B\) are equal. $$A=\left[\begin{array}{rrr}1 & x+y & 3 \\\4 & -1 & 6 \\\3 & 7 & -2\end{array}\right]$$ $$B=\left[\begin{array}{rrr}1 & 2 & 3 \\\4 & -1 & 6 \\\3 & y & -2\end{array}\right]$$

Graph the solution set to the inequality. $$ x+y>-3 $$

Find the specified minor and cofactor for \(A\). $$ M_{23} \text { and } A_{23} \text { if } A=\left[\begin{array}{llr} 1 & 2 & -1 \\ 4 & 6 & -3 \\ 2 & 3 & 9 \end{array}\right] $$

Determine if \(B\) is the inverse matrix of \(A\) by calculating \(A B\) and \(B A\) $$ A=\left[\begin{array}{rrr} 1 & -1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & -1 \\ 0 & 1 & 0 \\ -1 & -2 & 1 \end{array}\right] $$

Represent the linear system by an augmented matrix, and state the dimension of the matrix. $$ \begin{array}{rr} x+2 y-z= & 2 \\ -2 x+y-2 z= & -3 \\ 7 x+y-z= & 7 \end{array} $$

Determine whether each ordered triple is a solution to the system of linear equations. $$ \begin{array}{l} (5,2,2),(2,-1,1) \\ 2 x-3 y+3 z=10 \\ x-2 y-3 z=1 \\ 4 x-y+z=10 \end{array} $$

Evaluate the expression for the given \(f(x, y)\). $$ f(5,-2) \text { if } f(x, y)=6 y-\frac{1}{2} x $$

Graph the solution set to the inequality. $$ 2 x+y>4 $$

Find the specified minor and cofactor for \(A\). $$ M_{22} \text { and } A_{22} \text { if } A=\left[\begin{array}{rrr} 7 & -8 & 1 \\ 3 & -5 & 2 \\ 1 & 0 & -2 \end{array}\right] $$

Find the value of the constant \(k\) in \(A^{-1}\). $$ A=\left[\begin{array}{ll} 1 & 1 \\ 1 & 2 \end{array}\right], \quad A^{-1}=\left[\begin{array}{rr} 2 & -1 \\ -1 & k \end{array}\right] $$

Write the system of linear equations that the augmented matrix represents. $$ \left[\begin{array}{ll|l} 3 & 2 & 4 \\ 0 & 1 & 5 \end{array}\right] $$

Evaluate the expression for the given \(f(x, y)\). $$ f\left(\frac{1}{2},-\frac{7}{4}\right) \text { if } f(x, y)=\frac{2 x}{y+3} $$

Determine whether each ordered triple is a solution to the system of linear equations. $$ \begin{array}{c} \left(-\frac{5}{11}, \frac{20}{11},-2\right),(1,2,-1) \\ x+3 y-2 z=9 \\ -3 x+2 y+4 z=-3 \\ -2 x+5 y+2 z=6 \end{array} $$

Graph the solution set to the inequality. $$ 2 x+3 y \leq 6 $$

Find the specified minor and cofactor for \(A\). $$ M_{31} \text { and } A_{31} \text { if } A=\left[\begin{array}{rrr} 0 & 0 & -1 \\ 6 & -7 & 1 \\ 8 & -9 & -1 \end{array}\right] $$

Find the value of the constant \(k\) in \(A^{-1}\). A=\left[\begin{array}{rr} -2 & 2 \\ 1 & -2 \end{array}\right], \quad A^{-1}=\left[\begin{array}{rr} -1 & k \\ -0.5 & -1 \end{array}\right]

Write the system of linear equations that the augmented matrix represents. $$ \left[\begin{array}{rr|r} -2 & 1 & 5 \\ 7 & 9 & 2 \end{array}\right] $$

Evaluate the expression for the given \(f(x, y)\). $$ f(0.2,0.5) \text { if } f(x, y)=\frac{5 x}{2 y+1} $$