Chapter 8

Find the partial fraction decomposition. \(\frac{8 x-1}{(x-2)(x+3)}\)

Find all the minors and cofactors of the elements in the matrix. $$\left[\begin{array}{rr} 7 & -1 \\ 5 & 0 \end{array}\right]$$

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{array}\right|=-\left|\begin{array}{lll} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right|$$

Use matrices to solve the system. $$\left\\{\begin{array}{rr} x-2 y-3 z= & -1 \\ 2 x+y+z= & 6 \\ x+3 y-2 z= & 13 \end{array}\right.$$

Exer. \(1-2:\) Show that \(B\) is the inverse of \(A\) $$A=\left[\begin{array}{ll} 5 & 7 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -7 \\ -2 & 5 \end{array}\right]$$

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{rr} 5 & -2 \\ 1 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 4 & 1 \\ -3 & 2 \end{array}\right]$$

Solve the system. $$\left\\{\begin{aligned} 2 x+3 y &=2 \\ x-2 y &=8 \end{aligned}\right.$$

Sketch the graph of the Inequality. $$3 x-2 y<6$$

Use the method of substitution to solve the system. $$\left\\{\begin{array}{l} y=x^{2}-4 \\ y=2 x-1 \end{array}\right.$$

Find the partial fraction decomposition. \(\frac{x-29}{(x-4)(x+1)}\)

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{rr} 3 & 0 \\ -1 & 2 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -4 \\ 1 & 1 \end{array}\right]$$

Find all the minors and cofactors of the elements in the matrix. $$\left[\begin{array}{rr} -6 & 4 \\ 3 & 2 \end{array}\right]$$

Use matrices to solve the system. $$ \left\\{\begin{array}{rr} x+3 y-z= & -3 \\ 3 x-y+2 z= & 1 \\ 2 x-y+z= & -1 \end{array}\right. $$

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{array}\right|=-\left|\begin{array}{lll} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 1 \end{array}\right|$$

Exer. \(1-2:\) Show that \(B\) is the inverse of \(A\) $$A=\left[\begin{array}{rr} 8 & -5 \\ -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{ll} 2 & 5 \\ 3 & 8 \end{array}\right]$$

Solve the system. $$\left\\{\begin{array}{l} 4 x+5 y=13 \\ 3 x+y=-4 \end{array}\right.$$

Sketch the graph of the Inequality. $$4 x+3 y<12$$

Use the method of substitution to solve the system. $$\left\\{\begin{aligned} y &=x^{2}+1 \\ x+y &=3 \end{aligned}\right.$$

Find the partial fraction decomposition. \(\frac{x+34}{x^{2}-4 x-12}\)

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{rr} 6 & -1 \\ 2 & 0 \\ -3 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & 1 \\ -1 & 5 \\ 6 & 0 \end{array}\right]$$

Find all the minors and cofactors of the elements in the matrix. $$\left[\begin{array}{rrr} 2 & 4 & -1 \\ 0 & 3 & 2 \\ -5 & 7 & 0 \end{array}\right]$$

Use matrices to solve the system. $$\left\\{\begin{array}{rr} 5 x+2 y-z= & -7 \\ x-2 y+2 z= & 0 \\ 3 y+z= & 17 \end{array}\right.$$

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 1 & 0 & 1 \\ 2 & 1 & 0 \\ 1 & 1 & 2 \end{array}\right|=\left|\begin{array}{lll} 1 & 0 & 1 \\ 2 & 1 & 0 \\ 0 & 1 & 1 \end{array}\right|$$

Exer. \(3-12:\) Find the inverse of the matrix if it exists. $$\left[\begin{array}{rr} 2 & -4 \\ 1 & 3 \end{array}\right]$$

Solve the system. $$\left\\{\begin{array}{l} 2 x+5 y=16 \\ 3 x-7 y=24 \end{array}\right.$$

Sketch the graph of the Inequality. $$2 x+3 y \geq 2 y+1$$

Sketch the region \(R\) determined by the given constraints, and label its vertices. Find the maximum value of \(C\) on \(R\) $$\begin{array}{lll} C=3 x+y ; & x \geq 0, y \geq 0 \\ 3 x-4 y \geq-12, & 3 x+2 y \leq 24, & 3 x-y \leq 15 \end{array}$$

Use the method of substitution to solve the system. $$\left\\{\begin{aligned} y^{2} &=1-x \\ x+2 y &=1 \end{aligned}\right.$$

Find the partial fraction decomposition. \(\frac{5 x-12}{x^{2}-4 x}\)

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{rrr} 0 & -2 & 7 \\ 5 & 4 & -3 \end{array}\right], \quad B=\left[\begin{array}{lll} 8 & 4 & 0 \\ 0 & 1 & 4 \end{array}\right]$$

Find all the minors and cofactors of the elements in the matrix. $$\left[\begin{array}{rrr} 5 & -2 & 1 \\ 4 & 7 & 0 \\ -3 & 4 & -1 \end{array}\right]$$

Use matrices to solve the system. $$\left\\{\begin{aligned} 4 x-y+3 z &=6 \\ -8 x+3 y-5 z &=-6 \\ 5 x-4 y &=-9 \end{aligned}\right.$$

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{array}\right|=\left|\begin{array}{lll} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 2 & 1 & 1 \end{array}\right|$$

Exer. \(3-12:\) Find the inverse of the matrix if it exists. $$\left[\begin{array}{ll} 3 & 2 \\ 4 & 5 \end{array}\right]$$

Solve the system. $$\left\\{\begin{array}{l} 7 x-8 y=9 \\ 4 x+3 y=-10 \end{array}\right.$$

Sketch the region \(R\) determined by the given constraints, and label its vertices. Find the maximum value of \(C\) on \(R\) $$\begin{array}{l} C=4 x-2 y \\ x-2 y \geq-8, \quad 7 x-2 y \leq 28, \quad x+y \geq 4 \end{array}$$

Sketch the graph of the Inequality. $$2 x-y>3$$

Use the method of substitution to solve the system. $$\left\\{\begin{aligned} y^{2} &=x \\ x+2 y+3 &=0 \end{aligned}\right.$$

Find the partial fraction decomposition. \(\frac{4 x^{2}-15 x-1}{(x-1)(x+2)(x-3)}\)

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{lll} 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{lll} 7 & 0 & -5 \end{array}\right]$$

Use matrices to solve the system. $$\left\\{\begin{array}{rr} 2 x+6 y-4 z= & 1 \\ x+3 y-2 z= & 4 \\ 2 x+y-3 z= & -7 \end{array}\right.$$

Without expanding, explain why the statement is true. $$\left|\begin{array}{lll} 2 & 4 & 2 \\ 1 & 2 & 4 \\ 2 & 6 & 4 \end{array}\right|=4\left|\begin{array}{lll} 1 & 2 & 1 \\ 1 & 2 & 4 \\ 1 & 3 & 2 \end{array}\right|$$

Exer. \(3-12:\) Find the inverse of the matrix if it exists. $$\left[\begin{array}{ll} 2 & 4 \\ 4 & 8 \end{array}\right]$$

Solve the system. $$\left\\{\begin{aligned} 3 r+4 s &=3 \\ r-2 s &=-4 \end{aligned}\right.$$

Sketch the region \(R\) determined by the given constraints, and label its vertices. Find the minimum value of \(C\) on \(R\). $$\begin{aligned} &C=3 x+6 y ; \quad x \geq 0, y \geq 0\\\ &2 x+3 y \geq 12, \quad 2 x+5 y \geq 16 \end{aligned}$$

Sketch the graph of the Inequality. $$y+2

Use the method of substitution to solve the system. $$\left\\{\begin{aligned} 2 y &=x^{2} \\ y &=4 x^{3} \end{aligned}\right.$$

Find the partial fraction decomposition. \(\frac{x^{2}+19 x+20}{x(x+2)(x-5)}\)

Find, if possible, \(A+B, A-B, 2 A,\) and \(-3 B\). $$A=\left[\begin{array}{r} 7 \\ -16 \end{array}\right], \quad B=\left[\begin{array}{r} -11 \\ 9 \end{array}\right]$$

Use matrices to solve the system. $$\left\\{\begin{aligned} x+3 y-3 z &=-5 \\ 2 x-y+z &=-3 \\ -6 x+3 y-3 z &=4 \end{aligned}\right.$$