Chapter 8

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{7}{(x-3)(x+5)} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(B=\left[\begin{array}{rr}12 & -7 \\ -5 & 3\end{array}\right]\)

Find the inverse of the matrix or state that the matrix is not invertible. $$ A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right] $$

For each pair of matrices \(A\) and \(B\) in Exercises \(1-7,\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{rr} 2 & -3 \\ 1 & 4 \end{array}\right], B=\left[\begin{array}{rr} 5 & -2 \\ 4 & 8 \end{array}\right] $$

State whether the given matrix is in reduced row echelon form, row echelon form only or in neither of those forms. $$ \left[\begin{array}{ll|l} 1 & 0 & 3 \\ 0 & 1 & 3 \end{array}\right] $$

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{array}{r} x+2 y=5 \\ x=6 \end{array}\right. $$

Solve the given system of nonlinear equations. Sketch the graph of both equations on the same set of axes to verify the solution set. $$ \left\\{\begin{array}{r} x^{2}+y^{2}=4 \\ x^{2}-y=5 \end{array}\right. $$

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{5 x+4}{x(x-2)(2-x)} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(C=\left[\begin{array}{rr}6 & 15 \\ 14 & 35\end{array}\right]\)

Find the inverse of the matrix or state that the matrix is not invertible. $$ B=\left[\begin{array}{rr} 12 & -7 \\ -5 & 3 \end{array}\right] $$

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{ll} -1 & 5 \\ -3 & 6 \end{array}\right], B=\left[\begin{array}{rr} 2 & 10 \\ -7 & 1 \end{array}\right] $$

State whether the given matrix is in reduced row echelon form, row echelon form only or in neither of those forms. $$ \left[\begin{array}{rrr|r} 3 & -1 & 1 & 3 \\ 2 & -4 & 3 & 16 \\ 1 & -1 & 1 & 5 \end{array}\right] $$

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{aligned} 2 y-3 x &=1 \\ y &=-3 \end{aligned}\right. $$

Solve the given system of nonlinear equations. Sketch the graph of both equations on the same set of axes to verify the solution set. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=16 \\ 16 x^{2}+4 y^{2} &=64 \end{aligned}\right. $$

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{m}{(7 x-6)\left(x^{2}+9\right)} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(Q=\left[\begin{array}{ll}x & x^{2} \\ 1 & 2 x\end{array}\right]\)

Find the inverse of the matrix or state that the matrix is not invertible. $$ C=\left[\begin{array}{rr} 6 & 15 \\ 14 & 35 \end{array}\right] $$

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{rr} -1 & 3 \\ 5 & 2 \end{array}\right], B=\left[\begin{array}{rrr} 7 & 0 & 8 \\ -3 & 1 & 4 \end{array}\right] $$

State whether the given matrix is in reduced row echelon form, row echelon form only or in neither of those forms. $$ \left[\begin{array}{lll|l} 1 & 1 & 4 & 3 \\ 0 & 1 & 3 & 6 \\ 0 & 0 & 0 & 1 \end{array}\right] $$

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{array}{l} \frac{x+2 y}{4}=-5 \\ \frac{3 x-y}{2}=1 \end{array}\right. $$

Solve the given system of nonlinear equations. Sketch the graph of both equations on the same set of axes to verify the solution set. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=16 \\ 9 x^{2}-16 y^{2} &=144 \end{aligned}\right. $$

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{a x^{2}+b x+c}{x^{3}(5 x+9)\left(3 x^{2}+7 x+9\right)} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(L=\left[\begin{array}{cc}\frac{1}{x^{3}} & \frac{\ln (x)}{x^{3}} \\\ -\frac{3}{x^{4}} & \frac{1-3 \ln (x)}{x^{4}}\end{array}\right]\)

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{ll} 2 & 4 \\ 6 & 8 \end{array}\right], B=\left[\begin{array}{rrr} -1 & 3 & -5 \\ 7 & -9 & 11 \end{array}\right] $$

State whether the given matrix is in reduced row echelon form, row echelon form only or in neither of those forms. $$ \left[\begin{array}{lll|l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] $$

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{array}{l} \frac{2}{3} x-\frac{1}{5} y=3 \\ \frac{1}{2} x+\frac{3}{4} y=1 \end{array}\right. $$

Solve the given system of nonlinear equations. Sketch the graph of both equations on the same set of axes to verify the solution set. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=16 \\ \frac{1}{9} y^{2}-\frac{1}{16} x^{2} &=1 \end{aligned}\right. $$

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{\text { A polynomial of degree }<9}{(x+4)^{5}\left(x^{2}+1\right)^{2}} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(F=\left[\begin{array}{rrr}4 & 6 & -3 \\ 3 & 4 & -3 \\ 1 & 2 & 6\end{array}\right]\)

Find the inverse of the matrix or state that the matrix is not invertible. $$ E=\left[\begin{array}{rrr} 3 & 0 & 4 \\ 2 & -1 & 3 \\ -3 & 2 & -5 \end{array}\right] $$

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{l} 7 \\ 8 \\ 9 \end{array}\right], B=\left[\begin{array}{lll} 1 & 2 & 3 \end{array}\right] $$

State whether the given matrix is in reduced row echelon form, row echelon form only or in neither of those forms. $$ \left[\begin{array}{llll|l} 1 & 0 & 4 & 3 & 0 \\ 0 & 1 & 3 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right] $$

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{aligned} \frac{1}{2} x-\frac{1}{3} y &=-1 \\ 2 y-3 x &=6 \end{aligned}\right. $$

Solve the given system of nonlinear equations. Sketch the graph of both equations on the same set of axes to verify the solution set. $$ \left\\{\begin{aligned} x^{2}+y^{2} &=16 \\ x-y &=2 \end{aligned}\right. $$

In Exercises \(1-6\), find only the form needed to begin the process of partial fraction decomposition. Do not create the system of linear equations or attempt to find the actual decomposition. $$ \frac{\text { A polynomial of degree }<7}{x(4 x-1)^{2}\left(x^{2}+5\right)\left(9 x^{2}+16\right)} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(G=\left[\begin{array}{llr}1 & 2 & 3 \\ 2 & 3 & 11 \\ 3 & 4 & 19\end{array}\right]\)

Find the inverse of the matrix or state that the matrix is not invertible. $$ F=\left[\begin{array}{rrr} 4 & 6 & -3 \\ 3 & 4 & -3 \\ 1 & 2 & 6 \end{array}\right] $$

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{rr} 1 & -2 \\ -3 & 4 \\ 5 & -6 \end{array}\right], B=\left[\begin{array}{lll} -5 & 1 & 8 \end{array}\right] $$

State whether the given matrix is in reduced row echelon form, row echelon form only or in neither of those forms. $$ \left[\begin{array}{lll|l} 1 & 1 & 4 & 3 \\ 0 & 1 & 3 & 6 \end{array}\right] $$

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{array}{r} x+4 y=6 \\ \frac{1}{12} x+\frac{1}{3} y=\frac{1}{2} \end{array}\right. $$

Solve the given system of nonlinear equations. Use a graph to help you avoid any potential extraneous solutions. $$ \left\\{\begin{aligned} x^{2}-y^{2} &=1 \\ x^{2}+4 y^{2} &=4 \end{aligned}\right. $$

In Exercises 7 - 18 , find the partial fraction decomposition of the following rational expressions. $$ \frac{2 x}{x^{2}-1} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(V=\left[\begin{array}{rrr}i & j & k \\ -1 & 0 & 5 \\ 9 & -4 & -2\end{array}\right]\)

Find the inverse of the matrix or state that the matrix is not invertible. $$ G=\left[\begin{array}{rrr} 1 & 2 & 3 \\ 2 & 3 & 11 \\ 3 & 4 & 19 \end{array}\right] $$

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{rrr} 2 & -3 & 5 \\ 3 & 1 & -2 \\ -7 & 1 & -1 \end{array}\right], B=\left[\begin{array}{rrr} 1 & 2 & 1 \\ 17 & 33 & 19 \\ 10 & 19 & 11 \end{array}\right] $$

The following matrices are in reduced row echelon form. Determine the solution of the corresponding system of linear equations or state that the system is inconsistent. $$ \left[\begin{array}{rr|r} 1 & 0 & -2 \\ 0 & 1 & 7 \end{array}\right] $$

In Exercises \(1-8\), take a trip down memory lane and solve the given system using substitution and/or elimination. Classify each system as consistent independent, consistent dependent, or inconsistent. Check your answers both algebraically and graphically. $$ \left\\{\begin{aligned} 3 y-\frac{3}{2} x &=-\frac{15}{2} \\ \frac{1}{2} x-y &=\frac{3}{2} \end{aligned}\right. $$

Solve the given system of nonlinear equations. Use a graph to help you avoid any potential extraneous solutions. $$ \left\\{\begin{aligned} \sqrt{x+1}-y &=0 \\ x^{2}+4 y^{2} &=4 \end{aligned}\right. $$

In Exercises 7 - 18 , find the partial fraction decomposition of the following rational expressions. $$ \frac{-7 x+43}{3 x^{2}+19 x-14} $$

Compute the determinant of the given matrix. (Some of these matrices appeared in Exercises \(1-8\) in Section 8.4.) \(H=\left[\begin{array}{rrrr}1 & 0 & -3 & 0 \\ 2 & -2 & 8 & 7 \\ -5 & 0 & 16 & 0 \\ 1 & 0 & 4 & 1\end{array}\right]\)