Exam 7: Linear Systems and Matrices

Given: $A = \left[ \begin{array} { l l l } 4 & 2 & 5 \\1 & 2 & 1\end{array} \right] , B = \left[ \begin{array} { r r r } 5 & 5 & - 1 \\4 & 4 & 1\end{array} \right] , C = \left[ \begin{array} { r r } - 2 & - 1 \\3 & 1 \\- 2 & - 3\end{array} \right]$ determine $C A - C B$ .

Solve the system of equations graphically or algebraically. $\left\{ \begin{array} { l } y = 13 x + 24 \\y = \sqrt { 3 x + 576 }\end{array} \right.$

Solve using any method. $\left\{ \begin{aligned}- 7 x - 9 y & = - 6 \\y & = x - 3\end{aligned} \right.$

Determine which one of the ordered triples below is a solution of the given system of equations. $\left\{ \begin{array} { l } 8 x + 6 y - 4 z = 74 \\x - 5 y + 8 z = - 102 \\5 x - 3 y + 2 z = - 46\end{array} \right.$

If possible, find $4 A + 2 B$ $A = \left[ \begin{array} { c c c } 0 & - 4 & 4 \\1 & 0 & - 3\end{array} \right] , B = \left[ \begin{array} { c c c } - 3 & 2 & 5 \\- 5 & 8 & - 2\end{array} \right]$

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

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

Evaluate the determinant $\left| \begin{array} { c c } 7 x & x \ln x \\ 7 & 2 + \ln x \end{array} \right|$ in which the entries are functions.

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

Solve the system by the method of substitution. $\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 25 \\3 x - 4 y = 0\end{array} \right.$

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

Solve the system of linear equations. $\left\{ \begin{aligned}x + y + z & = - 6 \\x - 6 y - 7 z & = 76 \\8 y - 6 z & = - 18\end{aligned} \right.$

Evaluate the expression. $\left[ \begin{array} { c } 2 \\5 \\- 3\end{array} \right] \left( \left[ \begin{array} { l l } 6 & 4\end{array} \right] + \left[ \begin{array} { l l } - 8 & 1\end{array} \right] \right)$

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

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

Match the system of linear equations with its graph. $\left\{ \begin{array} { r } 3 x + 6 y - 12 = 0 \\15 x - 5 y - 10 = 0\end{array} \right.$

Solve using any method. $\left\{ \begin{aligned}7 x - 9 y & = 3 \\y & = x - 4\end{aligned} \right.$

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, $s = 273$ feet At $t = 2$ seconds, $s = 241$ feet At $t = 3$ seconds, $s = 177$ feet

Use the matrix capabilities of a graphing utility to reduce the augmented matrix corresponding to the system of equations, and solve the system. $\left\{ \begin{aligned}x - 7 y - 3 z & = 64 \\3 x + 7 y + 2 z & = - 92 \\- 2 x + 14 y + 6 z & = 64 \\- 8 x + 2 y + 2 z & = 42\end{aligned} \right.$

Use the matrix capabilities of a graphing utility to find the inverse of the matrix $\frac { 1 } { 6 } \left[ \begin{array} { c c c } - 4 & - 5 & 3 \\ - 4 & - 8 & 3 \\ 1 & 2 & 0 \end{array} \right]$ (if it exists).