Exam 6: Matrices and Determinants

Write a system of linear equations in three variables, and then use matrices to solve the system. -Ron attends a cocktail party (with his graphing calculator in his pocket). He wants to limit his food intake to 131 g protein, 107 g fat, and 165 g carbohydrate. According to the health conscious hostess, the Marinated mushroom caps have 3 g protein, 5 g fat, and 9 g carbohydrate; the spicy meatballs have 14 g Protein, 7 g fat, and 15 g carbohydrate; and the deviled eggs have 13 g protein, 15 g fat, and 6 g Carbohydrate. How many of each snack can he eat to obtain his goal?

Use Matrices and Gauss-Jordan Elimination to Solve Systems Solve the system of equations using matrices. Use Gauss-Jordan elimination. - 6x-7y-z= 9 x+4y-7z= -12 -2x+y+z= -5

Use Cramer's rule to solve the system. - 4x+4z =40 -7x+5y+6z =64 -2x-2y =-16

Use Cramer's rule to solve the system. - -2x-4y-4z=-62 -4y+3z=-7 -4x-4z=-40

Understand What is Meant by Equal Matrices Find values for the variables so that the matrices are equal. - $\left[ \begin{array} { l } x \\9\end{array} \right] = \left[ \begin{array} { l } 1 \\y\end{array} \right]$

Use Inverses to Solve Matrix Equations Write the linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix. - 5x+2y-2z=37 4x+3y+5z=22 7x-2y+6z=15

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { l l } - 5 & 1 \\ - 7 & 1 \end{array} \right] , \quad B = \left[ \begin{array} { l } \frac { 1 } { 2 } - \frac { 1 } { 2 } \\ \frac { 7 } { 2 } - \frac { 5 } { 2 } \end{array} \right]$

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. - $\left[ \begin{array} { r r r | r } 6 & 9 & 6 & - 2 \\ 2 & 0 & 7 & 4 \\ 6 & 8 & 0 & 2 \end{array} \right]$

Evaluate the determinant. - $\left| \begin{array} { l l } 5 & 7 \\ 3 & 8 \end{array} \right|$

Find the product AB, if possible. - $A = \left[ \begin{array} { r r r } 4 & 4 & - 6 \\- 3 & - 9 & 5 \\9 & 9 & - 9\end{array} \right] , B = \left[ \begin{array} { r r r } - 4 & - 3 & 4 \\6 & - 4 & - 7 \\5 & - 2 & 1\end{array} \right]$

Solve the matrix equation for X. -Let $A = \left[ \begin{array} { r r } 3 & - 3 \\ - 4 & 0 \\ 4 & - 5 \end{array} \right]$ and $B = \left[ \begin{array} { r r } 4 & 0 \\ 0 & - 2 \\ 3 & - 5 \end{array} \right]$

Use Inverses to Solve Matrix Equations Write the linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix. - $9 x - 2 y = 51$ $6 y = 36$

Use Matrices and Gauss-Jordan Elimination to Solve Systems Solve the system of equations using matrices. Use Gauss-Jordan elimination. - 3x+5y+2w =-12 2x+6z-w =-5 -2y+3z-3w =-3 -x+2y+4z+w =-2

Find the products AB and BA to determine whether B is the multiplicative inverse of A. - $A = \left[ \begin{array} { r r r } 2 & - 1 & 0 \\- 1 & 1 & - 2 \\1 & 0 & - 1\end{array} \right] , \quad B = \left[ \begin{array} { r r r } 1 & - 1 & 2 \\- 3 & - 2 & 4 \\- 1 & 1 & 1\end{array} \right]$

Use Cramer's rule to determine if the system is inconsistent system or contains dependent equations. - 4x+y=12 8x+2y=24

Solve the problem. -Let $A = \left[ \begin{array} { r r r } 2 & 2 & 3 \\ 4 & 0 & - 3 \\ - 4 & 6 & 9 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 6 & - 2 & 3 \\ - 4 & 0 & 1 \\ 3 & 9 & - 6 \end{array} \right]$ . Find $A = B$ .

Apply Gaussian Elimination to Systems Without Unique Solutions Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists. - 3x-2y+2z-w=2 4x+y+z+6w=8 -3x+2y-2z+w=5 5x+3z-2w=1

Solve the problem. -Let $A = \left[ \begin{array} { r r r } - 5 & 4 & 4 \\ 4 & 7 & 8 \\ 5 & - 6 & 3 \end{array} \right]$ and $B = \left[ \begin{array} { r r r } 8 & 7 & - 2 \\ - 5 & - 9 & - 8 \\ - 6 & 7 & - 2 \end{array} \right]$ . Find $- 4 A - 3 B$ .

Solve the matrix equation for X. -Let $\mathrm { A } = \left[ \begin{array} { r r } 1 & 3 \\ - 1 & - 3 \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { r r } - 1 & - 3 \\ 1 & - 4 \end{array} \right] ; \quad \mathrm { X } + \mathrm { A } = \mathrm { B }$

Write the system of linear equations represented by the augmented matrix. Use x, y, z, and, if necessary, w for the variables. - $\left[ \begin{array} { r r r r | r } 6 & 1 & 0 & 8 & - 6 \\- 1 & 3 & 1 & 0 & 7 \\7 & 0 & 0 & 5 & - 4 \\0 & 4 & 0 & - 4 & 10\end{array} \right]$