Chapter 6

Determine whether each ordered triple is a solution to the system of linear equations. $$ \begin{aligned} (1,2,3),(11,16,-3) & \\ 4 x-2 y+2 z &=\\\ 2 x-4 y-6 z &=-24 \\ -3 x+3 y+2 z &=9 \end{aligned} $$

Let \(a_{i j}\) and \(b_{i j}\) be general elements for the given matrices \(A\) and \(B\). (a) Identify \(a_{12}, b_{32},\) and \(b_{22}\) (b) Compute \(a_{11} b_{11}+a_{12} b_{21}+a_{13} b_{31}\) (c) If possible, find a value for \(x\) that makes \(A=B\). $$A=\left[\begin{array}{rrr}1 & 3 & -4 \\\3 & 0 & 7 \\\x & 1 & -1\end{array}\right]$$ $$B=\left[\begin{array}{rrr}1 & x & -4 \\\3 & 0 & 7 \\\3 & 1 & -1\end{array}\right]$$

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

Let \(A\) be the given matrix. Find det \(A\) by expanding about the first column. State whether \(A^{-1}\) exists. $$ \left[\begin{array}{rrr} 1 & 4 & -7 \\ 0 & 2 & -3 \\ 0 & -1 & 3 \end{array}\right] $$

Find the value of the constant \(k\) in \(A^{-1}\). A=\left[\begin{array}{rr} 1 & 3 \\ -1 & -5 \end{array}\right], \quad A^{-1}=\left[\begin{array}{rr} k & 1.5 \\ -0.5 & -0.5 \end{array}\right]

Write the system of linear equations that the augmented matrix represents. $$ \left[\begin{array}{rrr|r} 3 & 1 & 4 & 0 \\ 0 & 5 & 8 & -1 \\ 0 & 0 & -7 & 1 \end{array}\right] $$

Write a symbolic representation for \(f(x, y)\) if the function \(f\) computes the following quantity. The sum of \(y\) and twice \(x\).

If possible, solve the system. $$ \begin{array}{r} x+y+z=6 \\ -x+2 y+z=6 \\ y+z=5 \end{array} $$

Let \(a_{i j}\) and \(b_{i j}\) be general elements for the given matrices \(A\) and \(B\). (a) Identify \(a_{12}, b_{32},\) and \(b_{22}\) (b) Compute \(a_{11} b_{11}+a_{12} b_{21}+a_{13} b_{31}\) (c) If possible, find a value for \(x\) that makes \(A=B\). $$A=\left[\begin{array}{rrr}0 & -1 & 6 \\\2 & x & -1 \\\9 & -2 & 1\end{array}\right]$$ $$B=\left[\begin{array}{rrr}0 & -1 & x \\\2 & 6 & -1 \\\7 & -2 & 1\end{array}\right]$$

Graph the solution set to the inequality. $$ x^{2}+y^{2} \leq 1 $$

Let \(A\) be the given matrix. Find det \(A\) by expanding about the first column. State whether \(A^{-1}\) exists. $$ \left[\begin{array}{rrr} 0 & 2 & 8 \\ -1 & 3 & 5 \\ 0 & 4 & 1 \end{array}\right] $$

Find the value of the constant \(k\) in \(A^{-1}\). A=\left[\begin{array}{ll} -2 & 5 \\ -3 & 4 \end{array}\right], \quad A^{-1}=\left[\begin{array}{ll} \frac{4}{7} & -\frac{5}{7} \\ k & -\frac{2}{7} \end{array}\right]

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

Write a symbolic representation for \(f(x, y)\) if the function \(f\) computes the following quantity. The product of \(x^{2}\) and \(y^{2}\)

If possible, solve the system. $$ \begin{array}{rr} x-y+z= & -2 \\ x-2 y+z= & 0 \\ y-z= & 1 \end{array} $$

For the given matrices \(A\) and \(B\) find each of the following. (a) \(\boldsymbol{A}+\boldsymbol{B} \quad\) (b) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsymbol{c}) \boldsymbol{A}-\boldsymbol{B}\) $$A=\left[\begin{array}{rr}4 & -1 \\\\-1 & 4\end{array}\right]$$ $$B=\left[\begin{array}{rr}-1 & 4 \\\4 & -1\end{array}\right]$$

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

Let \(A\) be the given matrix. Find det \(A\) by expanding about the first column. State whether \(A^{-1}\) exists. $$ \left[\begin{array}{rrr} 5 & 1 & 6 \\ 0 & -2 & 0 \\ 0 & 4 & 0 \end{array}\right] $$

Predict the results of \(I_{n} A\) and \(A I_{n}\). Then verify your prediction. $$ I_{2}=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right], \quad A=\left[\begin{array}{rr} 1 & -2 \\ 4 & 3 \end{array}\right] $$

Is the matrix in row-echelon form? (a) \(\left[\begin{array}{ll|r}1 & 3 & 2 \\ 0 & 1 & -1\end{array}\right]\) (b) \(\left[\begin{array}{rrr|r}1 & 4 & -1 & 0 \\ 0 & -1 & 1 & 3 \\ 0 & 2 & 1 & 7\end{array}\right]\) (c) \(\left[\begin{array}{rrr|r}1 & 6 & -8 & 5 \\ 0 & 1 & 7 & 9 \\ 0 & 0 & 1 & 11\end{array}\right]\)

Write a symbolic representation for \(f(x, y)\) if the function \(f\) computes the following quantity. The product of \(x\) and \(y\) divided by \(1+x\).

If possible, solve the system. $$ \begin{aligned} x+2 y+3 z &=4 \\ 2 x+y+3 z &=5 \\ x-y+z &=2 \end{aligned} $$

For the given matrices \(A\) and \(B\) find each of the following. (a) \(\boldsymbol{A}+\boldsymbol{B} \quad\) (b) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsymbol{c}) \boldsymbol{A}-\boldsymbol{B}\) $$A=\left[\begin{array}{rr}2 & -4 \\\\-1 & \frac{1}{2} \\\3 & -2\end{array}\right]$$ $$B=\left[\begin{array}{rr}5 & 0 \\\3 & \frac{1}{2} \\\\-1 & 1\end{array}\right]$$

Graph the solution set to the inequality. $$ 2 x^{2}-y<1 $$

Let \(A\) be the given matrix. Find det \(A\) by expanding about the first column. State whether \(A^{-1}\) exists. $$ \left[\begin{array}{lll} 3 & 2 & 3 \\ 2 & 2 & 2 \\ 1 & 3 & 1 \end{array}\right] $$

Predict the results of \(I_{n} A\) and \(A I_{n}\). Then verify your prediction. $$ I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \quad A=\left[\begin{array}{rrr} 1 & -4 & 3 \\ 1 & 9 & 5 \\ 3 & -5 & 0 \end{array}\right] $$

Is the matrix in row-echelon form? (a) \(\left[\begin{array}{rr|r}1 & 3 & 2 \\ 0 & -1 & -1\end{array}\right]\) (b) \(\left[\begin{array}{rrr|r}1 & 3 & -1 & 8 \\ 0 & 1 & 5 & 3 \\ 0 & 0 & 0 & 0\end{array}\right]\) (c) \(\left[\begin{array}{rrr|r}0 & 0 & 1 & 1 \\ 0 & 1 & 7 & 9 \\ 1 & 2 & -1 & 11\end{array}\right]\)

Write a symbolic representation for \(f(x, y)\) if the function \(f\) computes the following quantity. The square root of the sum of \(x\) and \(y\).

If possible, solve the system. $$ \begin{aligned} x-y+z &=2 \\ 3 x-2 y+z &=-1 \\ x+y &=-3 \end{aligned} $$

For the given matrices \(A\) and \(B\) find each of the following. (a) \(\boldsymbol{A}+\boldsymbol{B} \quad\) (b) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsymbol{c}) \boldsymbol{A}-\boldsymbol{B}\) $$A=\left[\begin{array}{rrr}3 & 4 & -1 \\\0 & -3 & 2 \\\\-2 & 5 & 10\end{array}\right]$$ $$B=\left[\begin{array}{rrr}11 & 5 & -2 \\\4 & -7 & 12 \\\6 & 6 & 6\end{array}\right]$$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{lll} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array}\right] $$

Predict the results of \(I_{n} A\) and \(A I_{n}\). Then verify your prediction. $$ I_{3}=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right], \quad A=\left[\begin{array}{lll} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] $$

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rr|r} 1 & 2 & 3 \\ 0 & 1 & -1 \end{array}\right] $$

If possible, solve the system. $$ \begin{aligned} &3 x+y+z=0\\\ &4 x+2 y+z=1\\\ &2 x-2 y-z=2 \end{aligned} $$

For the given matrices \(A\) and \(B\) find each of the following. (a) \(\boldsymbol{A}+\boldsymbol{B} \quad\) (b) \(\boldsymbol{B}+\boldsymbol{A} \quad(\boldsymbol{c}) \boldsymbol{A}-\boldsymbol{B}\) $$A=\left[\begin{array}{rrrr}1 & 6 & 1 & -2 \\\0 & 1 & 3 & 5 \\\0 & 0 & 1 & -2\end{array}\right]$$ $$B=\left[\begin{array}{rrrr}1 & 0 & 0 & 9 \\\3 & 1 & 0 & 3 \\\\-1 & 4 & 1 & -2 \end{array}\right]$$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{lll} 0 & 0 & 2 \\ 0 & 3 & 0 \\ 5 & 0 & 0 \end{array}\right] $$

Predict the results of \(I_{n} A\) and \(A I_{n}\). Then verify your prediction. $$ I_{4}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right], \quad A=\left[\begin{array}{rrrr} 5 & -2 & 6 & -3 \\ 0 & 1 & 4 & -1 \\ -5 & 7 & 9 & 8 \\ 0 & 0 & 3 & 1 \end{array}\right] $$

If possible, solve the system. $$ \begin{array}{rr} -x-5 y+2 z= & 2 \\ x+y+2 z= & 2 \\ 3 x+y-4 z= & -10 \end{array} $$

If possible, find each of the following. (a) \(A+B\) (b) \(3 A\) (c) \(2 A-3 B\) $$A=\left[\begin{array}{rr}2 & -6 \\\3& 1\end{array}\right]$$ $$B=\left[\begin{array}{ll}-1 & 0 \\\\-2 & 3\end{array}\right]$$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{rrr} 0 & 0 & 0 \\ -8 & 3 & -9 \\ 15 & 5 & 9 \end{array}\right] $$

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rr|r} 1 & -1 & 2 \\ 0 & 1 & 0 \end{array}\right] $$

Solve the equation for \(x\) and then solve it for \(y .\) $$ x-y^{2}=5 $$

If possible, solve the system. $$ \begin{array}{r} x+3 y+z=6 \\ 3 x+y-z=6 \\ x-y-z=0 \end{array} $$

If possible, find each of the following. (a) \(A+B\) (b) \(3 A\) (c) \(2 A-3 B\) $$A=\left[\begin{array}{rrr}1 & -2 & 5 \\\3 & -4 & -1\end{array}\right]$$ $$B=\left[\begin{array}{rrr}0 & -1 & -5 \\\\-3 & 1 & 2\end{array}\right]$$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{rrr} 1 & 1 & 5 \\ -3 & -3 & 0 \\ 7 & 0 & 0 \end{array}\right] $$

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{rr} 1 & 0 \\ 1 & -1 \end{array}\right] $$

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rr|r} 1 & 4 & -2 \\ 0 & 1 & 3 \end{array}\right] $$

If possible, solve the system. $$ \begin{array}{lr} 2 x-y+2 z= 6 \\ -x+y+z= 0 \\ -x \quad \quad -3 z=-6 \end{array} $$

Graph the solution set to the system of inequalities. Use the graph to identify one solution. $$ \begin{array}{r} y \geq x^{2} \\ x+y \leq 6 \end{array} $$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{rrr} 3 & -1 & 2 \\ 0 & 5 & 7 \\ 1 & 0 & -1 \end{array}\right] $$