Exam 7: Linear Systems and Matrices

Perform the indicated row operations on the matrix. Show the final result. $\left[ \begin{array} { r r r } 1 & - 2 & - 3 \\2 & - 3 & - 1 \\1 & - 8 & - 33\end{array} \right]$ Add $- 2$ times $R _ { 1 }$ to $R _ { 2 }$ . Add $- 1$ times $R _ { 1 }$ to $R _ { 3 }$ .

Evaluate the expression. $\frac { 2 } { 9 } \left[ \begin{array} { l l l } 5 & - 2 & 1\end{array} \right] + \left[ \begin{array} { l l l } 4 & 8 & - 1\end{array} \right]$

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } 9 & - 6 & 24 \\ - 9 & 6 & - 18 \\ 3 & - 9 & 18 \end{array} \right]$ .

Write the partial fraction decomposition of the rational expression. $\frac { x ^ { 2 } } { x ^ { 4 } - 21 x ^ { 2 } - 100 }$

Solve for $X$ in the equation given. $3 X = 2 A - B , A = \left[ \begin{array} { c c } - 6 & 2 \\8 & - 5\end{array} \right] \text { and } B = \left[ \begin{array} { c c } - 3 & 10 \\7 & - 1\end{array} \right]$

Find the equilibrium point of the demand and supply equations. (The equilibrium point is the price $p$ and number of units $x$ that satisfy both the demand and supply equations.) Demand Supply $p = 130 - 0.09 x$ $p = 0.4 x - 115$

Determine whether the system of linear equations is consistent or inconsistent. $\left\{ \begin{array} { c } - 9 x + 6 y = - 1 \\- 81 x + 54 y = - 10\end{array} \right.$

Evaluate the expression. $\left[ \begin{array} { c c } 4 & - 3 \\- 9 & 1\end{array} \right] + \left[ \begin{array} { c c } 2 & 8 \\- 2 & - 4\end{array} \right] + \left[ \begin{array} { c c } - 2 & - 1 \\- 8 & 4\end{array} \right]$

Use a determinant to determine whether the points below are collinear. $( 3 , - 1 ) , ( 0 , - 3 ) , ( 27,15 )$

Solve the system by the method of substitution. $\left\{ \begin{array} { l } y = - 3 x \\y = x ^ { 3 } + 3 x ^ { 2 } + 3 x\end{array} \right.$

Evaluate the expression. $\left[ \begin{array} { c c } 5 & - 1 \\4 & - 4\end{array} \right] \left[ \begin{array} { c c } 0 & 3 \\- 3 & 0\end{array} \right] \left[ \begin{array} { c c } - 5 & 0 \\1 & 4\end{array} \right]$

Use a system of equations to find the specified equation that passes through the points. Solve the system using matrices. Parabola: $y = a x ^ { 2 } + b x + c$

Given: $A = \left[ \begin{array} { r r r } - 7 & - 4 & 2 \\ 4 & - 5 & - 8 \end{array} \right] , B = \left[ \begin{array} { r r r } - 4 & 1 & - 7 \\ 8 & - 8 & 3 \end{array} \right]$ and $c = - 6$ , determine $c A B ^ { 2 }$ .

Find the inverse of the matrix $\left[ \begin{array} { c c } 4 & - 5 \\ 1 & 5 \end{array} \right]$ (if it exists).

Solve system of equations by the method of substitution. $\left\{ \begin{array} { r } 4 x ^ { 3 } - 25 y = 0 \\4 x - y = 0\end{array} \right.$

Write the system of linear equations as a matrix equation $A X = B$ , and use Gauss-Jordan elimination on the augmented matrix $[ A \vdots B ]$ to solve for the matrix $X$ . $\left\{ \begin{array} { l } x _ { 1 } - 3 x _ { 2 } - 9 x _ { 3 } = 43 \\x _ { 1 } + 4 x _ { 2 } + 7 x _ { 3 } = - 51 \\7 x _ { 1 } - 7 x _ { 2 } + 3 x _ { 3 } = - 57\end{array} \right.$

Use a system of equations to find the cubic function $f ( x ) = a x ^ { 3 } + b x ^ { 2 } + c x + d$ that satisfies given equations. Solve the system using matrices. f(2) =-26 f(-2) =6 f(1) =0 f(-1) =-2

Solve for $x$ given the following equation involving a determinant. $\left| \begin{array} { c c } x + 1 & 7 \\- 1 & x + 9\end{array} \right| = 0$

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. $\left\{ \begin{aligned} x + y - 9 z & = - 67 \\ y + 5 z & = 31 \\ z & = 7 \end{aligned} \right.$ $\left\{ \begin{aligned} x - 9 y + 6 z & = 76 \\ y - 9 z & = - 67 \\ z & = 7 \end{aligned} \right.$

Solve the system by the method of elimination. $\left\{ \begin{array} { c } 4 x + 5 y = - 71 \\x - y = - 2\end{array} \right.$