Exam 7: Linear Systems and Matrices

Given: $A = \left[ \begin{array} { r r r } 8 & - 7 & 0 \\ - 5 & - 2 & 3 \end{array} \right] , B = \left[ \begin{array} { r r r } 3 & 0 & 7 \\ 4 & 9 & - 2 \end{array} \right] , c = 8$ and $d = - 3$ , determine $c A + d B$ .

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

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. $( - 3,9 ) , ( - 2,7 ) , ( - 1,7 )$

Solve system of equations by the method of substitution. $\left\{ \begin{array} { r } 15 x ^ { 3 } - 25 y = 0 \\15 x - y = 0\end{array} \right.$

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+9y+9z =-22 y-6z =-23 z =3 x+8y-5z =-63 y+4z =7 z =3

Determine which ordered pair is a solution of the system. $\left\{ \begin{aligned}x - 2 y ^ { 2 } & = - 7 \\- 5 x + 6 y & = 7\end{aligned} \right.$

Solve the system of equations algebraically. $\left\{ \begin{array} { r } x y + 289 = 0 \\4 x - 25 y - 340 = 0\end{array} \right.$

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+9y-2z =41 y-7z =46 z =-6 x+4y+2z =-3 y+5z =-26 z =-6

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

Solve the system of linear equations. $\left\{ \begin{array} { c } x + 4 y + 2 z = - 2 \\- 4 x + y - 4 z = 9 \\3 x - y + 2 z = - 3\end{array} \right.$

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

Solve the system of linear equations $\left\{ \begin{array} { l } 6 x _ { 1 } - 12 x _ { 2 } - 6 x _ { 3 } - 12 x _ { 4 } = 0 \\18 x _ { 1 } - 30 x _ { 2 } - 12 x _ { 3 } - 18 x _ { 4 } = - 3 \\12 x _ { 1 } - 30 x _ { 2 } - 12 x _ { 3 } - 30 x _ { 4 } = 2 \\- 6 x _ { 1 } + 24 x _ { 2 } + 24 x _ { 3 } + 66 x _ { 4 } = 0\end{array} \right.$ using the inverse matrix $\frac { 1 } { 6 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \\ - 10 & 3 & 0 & - 1 \\ - 29 & 7 & 3 & - 2 \\ 12 & - 3 & - 1 & 1 \end{array} \right]$ .

Find the equation of the circle $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$ that passes through the points $( 1,9 ) , ( - 4,4 ) , ( 6,4 )$ .

Solve for $x$ given the following equation involving a determinant. $\left| \begin{array} { c c } x + 2 & 5 \\- 1 & x + 8\end{array} \right| = 0$

Write the system of linear equations represented by the augmented matrix. Then use back-substitution to solve. (Use variables $x , y$ , and $z$ .) $\left[ \begin{array} { c c c : c } 1 & 4 & - 3 & - 23 \\0 & 1 & 4 & 19 \\0 & 0 & 1 & 5\end{array} \right]$

Find $| A B |$ , if $A = \left[ \begin{array} { r r r } - 3 & 4 & 5 \\- 3 & - 5 & 1 \\5 & - 5 & 1\end{array} \right] , B = \left[ \begin{array} { r r r } - 1 & - 2 & 5 \\4 & 1 & - 3 \\- 1 & - 1 & 4\end{array} \right]$

Evaluate the expression. $\left[ \begin{array} { c c } - 7 & - 3 \\6 & 5\end{array} \right] + \left[ \begin{array} { c c } - 6 & 7 \\- 1 & 2\end{array} \right] + \left[ \begin{array} { c c } 4 & 9 \\9 & 4\end{array} \right]$

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

An object moving vertically is at the given heights at the specified times. Find the position equation $s = \frac { 1 } { 2 } a t ^ { 2 } + v _ { o } t + s _ { o }$ for the object. At $t = 1$ second, $s = 234$ feet At $t = 2$ seconds, $s = 202$ feet At $t = 3$ seconds, $s = 138$ feet

Determine which one of the ordered triples below is a solution of the given system of equations. $\left\{ \begin{array} { l } 7 x - 9 y + 6 z = 1 \\9 x + 2 y - 3 z = 37 \\3 x - 4 y + 5 z = 7\end{array} \right.$