Exam 7: Linear Systems and Matrices

Given: $A = \left[ \begin{array} { r r r } - 6 & - 10 & 7 \\ - 2 & 4 & 3 \end{array} \right] , B = \left[ \begin{array} { r r r } - 1 & 7 & - 3 \\ 1 & 2 & - 2 \end{array} \right] , c = - 4$ and $d = 2$ , determine $c A + d B$ .

Determine the order of the matrix. $\left[ \begin{array} { c c c } - 4 & - 8 & 2 \\0 & - 7 & - 4\end{array} \right]$

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

Find $x$ and $y$ . $\left[ \begin{array} { l l } 1 & x \\y & 3\end{array} \right] = \left[ \begin{array} { c c } 1 & - 6 \\- 4 & 3\end{array} \right]$

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 $x$ and $y$ . $\left[ \begin{array} { c c c } x + 3 & 6 & 4 y \\5 & 2 x & - 1 \\8 & y - 9 & 2\end{array} \right] = \left[ \begin{array} { c c c } 9 x + 19 & 6 & - 16 \\5 & - 4 & - 1 \\8 & - 13 & 2\end{array} \right]$

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { l } 3 x - 6 y = - 8 \\9 x + 8 y = 5\end{array} \right.$

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

Solve the system of linear equations $\left\{ \begin{array} { l } 8 x _ { 1 } - 16 x _ { 2 } - 8 x _ { 3 } - 16 x _ { 4 } = 0 \\24 x _ { 1 } - 40 x _ { 2 } - 16 x _ { 3 } - 24 x _ { 4 } = - 9 \\16 x _ { 1 } - 40 x _ { 2 } - 16 x _ { 3 } - 40 x _ { 4 } = 6 \\- 8 x _ { 1 } + 32 x _ { 2 } + 32 x _ { 3 } + 88 x _ { 4 } = 0\end{array} \right.$ using the inverse matrix $\frac { 1 } { 8 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \\ - 10 & 3 & 0 & - 1 \\ - 29 & 7 & 3 & - 2 \\ 12 & - 3 & - 1 & 1 \end{array} \right]$ .

Solve the system of linear equations $\left\{ \begin{array} { l l } 4 x _ { 1 } - 8 x _ { 2 } - 4 x _ { 3 } - 8 x _ { 4 } & = 0 \\ 12 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 12 x _ { 4 } & = - 9 \\ 8 x _ { 1 } - 20 x _ { 2 } - 8 x _ { 3 } - 20 x _ { 4 } & = 6 \\ - 4 x _ { 1 } + 16 x _ { 2 } + 16 x _ { 3 } + 44 x _ { 4 } & = 0 \end{array} \right.$ using the inverse matrix $\frac { 1 } { 4 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \\ - 10 & 3 & 0 & - 1 \\ - 29 & 7 & 3 & - 2 \\ 12 & - 3 & - 1 & 1 \end{array} \right]$

Find the minor $M _ { 13 }$ and its cofactor $C _ { 13 }$ of the matrix $\left[ \begin{array} { c c c } - 3 & 2 & - 8 \\ 3 & - 2 & 6 \\ - 1 & 3 & - 6 \end{array} \right]$ .

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

Use a determinant to determine whether the points below are collinear. $( 3 , - 1 ) , ( 0 , - 3 ) , ( 24,13 )$

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 determinant of $\left[ \begin{array} { c c c } 0 & - 6 & 0 \\ - 9 & 6 & - 18 \\ 3 & 0 & 18 \end{array} \right]$ by the method of expansion by cofactors.

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $\left\{ \begin{array} { c } x + 4 y + 8 z = 5 \\7 x - 3 y - 4 z = - 64 \\- x - 4 y - 8 z = - 5 \\6 x + 5 y + 7 z = - 18\end{array} \right.$

Solve the system of linear equations. $\left\{ \begin{aligned}x + y + z & = - 11 \\x - 6 y - 3 z & = 13 \\6 y - 7 z & = 42\end{aligned} \right.$

Solve the system of linear equations $\left\{ \begin{array} { l l } - 8 x + 24 y + 8 z & = 5 \\ 16 x + 40 y & = 10 \\ 24 x + 8 y - 16 z = - 5 \end{array} \right.$ using an inverse

Use the matrix capabilities of a graphing utility to find $A B$ , if possible. $A = \left[ \begin{array} { c c c } - 7 & 5 & 2 \\0 & 1 & 8 \\2 & 8 & - 2\end{array} \right] , B = \left[ \begin{array} { c c c } 4 & 3 & - 9 \\- 4 & - 1 & - 3 \\9 & 8 & 1\end{array} \right]$

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=130-0.09x p=0.4x-115