Exam 8: Matrices and Determinants

Select the augmented matrix for the system of linear equations. $\left\{ \begin{aligned}x + 14 y - 12 z & = 12 \\5 x - 4 y + 5 z & = 0 \\12 x + y & = 9\end{aligned} \right.$

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{array} { c } 4 x + 9 y + z = - 17 \\6 x - 6 y - 8 z = 12 \\9 x + 8 y + z = - 43\end{array} \right.$

Find a value of y such that the triangle with the given vertices has an area of 40 square units. ​ ​(7, 0), ​(7, -5), ​(-9, y) ​

Identify the elementary row operation being performed to obtain the new row-equivalent matrix. Original Matrix New Row-Equivalent Matrix $\left[ \begin{array} { r r r } - 9 & - 8 & 6 \\- 8 & - 5 & - 3\end{array} \right]$ $\left[ \begin{array} { r r r } 7 & 2 & 12 \\- 8 & - 5 & - 3\end{array} \right]$

Find A - B. $A = \left[ \begin{array} { c c c } - 2 & 5 & 0 \\3 & - 4 & 4 \\6 & 6 & - 1 \\0 & 8 & - 2 \\- 4 & - 4 & 0\end{array} \right] , B = \left[ \begin{array} { c c c } - 4 & 8 & 2 \\4 & - 4 & - 4 \\11 & - 8 & - 4 \\2 & 2 & - 4 \\0 & 1 & - 1\end{array} \right]$

Find the inverse of the matrix. $\left[ \begin{array} { l l } 4 & 1 \\1 & 4\end{array} \right]$

Identify the elementary row operation being performed to obtain the new row-equivalent matrix. Original Matrix New Row-Equivalent Matrix $\left[ \begin{array} { c c c } - 4 & 6 & 1 \\4 & - 1 & - 9\end{array} \right]$ $\left[ \begin{array} { c c c } 20 & 0 & - 53 \\4 & - 1 & - 9\end{array} \right]$

A coffee manufacturer sells a 14-pound package that contains three flavors of coffee for $25.French vanilla coffee costs $5 per pound, hazelnut flavored coffee costs $5.50 per pound, and Swiss chocolate flavored coffee costs $6 per pound.The package contains the same amount of hazelnut as Swiss chocolate.Let f represent the number of pounds of French vanilla, h represent the number of pounds of hazelnut, and s represent the number of pounds of Swiss chocolate. Write a system of linear equations that represents the situation.

Use Cramer's Rule to solve the following system of linear equations: $\left\{ \begin{array} { r } 4 x - 8 z = 2 \\- 4 y + 12 z = 3 \\8 x + 20 z = 0\end{array} \right.$

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

Find the inverse of the matrix (if it exists). $\left[ \begin{array} { l l } 1 & 2 \\3 & 7\end{array} \right]$

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $\left\{ \begin{aligned}x - 3 z & = - 7 \\3 x + y - 2 z & = 2 \\2 x + 2 y + z & = 4\end{aligned} \right.$

If possible, find AB. $A = \left[ \begin{array} { c c } - 5 & 1 \\7 & 5 \\8 & 8\end{array} \right] , B = \left[ \begin{array} { c } - 4 \\8\end{array} \right]$

Use a determinant to find an equation of the line passing through the points. ​ (8, 8), (-4, -8) ​

Find all the cofactors of the matrix. $\left[ \begin{array} { c c } 6 & 7 \\4 & - 8\end{array} \right]$

Select the order for the following matrix. $\left[ \begin{array} { r r r } - 9 & 8 & 7 \\0 & - 5 & 3\end{array} \right]$

Use the inverse formula $A ^ { - 1 } = \frac { 1 } { a d - b c } \left[ \begin{array} { c c } d & - b \\- c & a\end{array} \right]$ to find the inverse of the matrix (if it exists). $\left[ \begin{array} { c c } - 7 & - 9 \\7 & 9\end{array} \right]$

Find the determinant of $\left[ \begin{array} { c c c } 0 & - 2 & 0 \\- 3 & 2 & - 6 \\1 & 0 & 6\end{array} \right]$ by the method of expansion by cofactors using column 3.

A florist is creating 10 centerpieces for the tables at a wedding reception.Roses cost $2.50 each, lilies cost $8 each, and irises cost $4 each.The customer has a budget of $300 allocated for the centerpieces and wants each centerpiece to contain 12 flowers, with twice as many roses as the number of irises and lilies combined. Write a system of linear equations that represents the situation.

Perform the sequence of row operations on the matrix.What did the operations accomplish $\left[ \begin{array} { c c c } 1 & 1 & - 7 \\4 & 5 & - 7 \\1 & 3 & 35\end{array} \right]$ Add -4 times R₁ to R₂, Add -1 times R₁ to R₃, Add -2 times R₂ to R₃, Add -1 times R₂ to R₁.