Exam 7: Linear Systems and Matrices

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

Solve the system of equations below using Gaussian elimination. 5x+4y+3z =90 x-2y+2z =60 x-y-z =-30

Find the inverse of the matrix $\left[ \begin{array} { c c } 8 & 16 \\ - 24 & - 40 \end{array} \right]$ .

Given: $A = \left[ \begin{array} { r r r } 4 & 2 & - 3 \\1 & - 2 & 2\end{array} \right] , B = \left[ \begin{array} { r r r } 0 & - 2 & 5 \\5 & 4 & 2\end{array} \right] , C = \left[ \begin{array} { r r } - 4 & 1 \\- 5 & - 1 \\- 3 & - 3\end{array} \right]$ determine $C A - C B$ .

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

Write the partial fraction decomposition of the rational expression. $\frac { 1 } { 100 x ^ { 2 } - 81 }$

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

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

Find the determinant of the matrix $\left[ \begin{array} { r r } - \frac { 1 } { 6 } & - \frac { 1 } { 2 } \\ - 1 & - \frac { 4 } { 5 } \end{array} \right]$ .

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

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 $\frac { 1 } { 8 } \left[ \begin{array} { c c c c } - 24 & 7 & 1 & - 2 \\ - 10 & 3 & 0 & - 1 \\ - 29 & 7 & 3 & - 2 \\ 12 & - 3 & - 1 & 1 \end{array} \right]$ .

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

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 matrix.

Determine whether the two systems of linear equations yield the same solutions. If so, find the solutions using matrices. x+5y-5z =42 y-9z =48 z =-5 \{x+8y-3z =31 y+5z =-22 z =-5

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

Find the inverse of the matrix $\left[ \begin{array} { c c } 6 & 12 \\ - 18 & - 30 \end{array} \right]$ .

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.