Exam 15: Systems of Equations: Matrices and Determinants

Use Cramer s rule to find the solution set for the system. $\left( \begin{array} { c l c c c } 4 x - 5 y + z & = & 15 \\- 5 x + y - z & = & - 14 \\x - 4 y - 9 z & = & 25\end{array} \right)$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent .

The matrix is the reduced echelon matrix for a system with variables x 1 , x 2 , x 3 , and x 4 . Find the solution set of the system. $\left[ \begin{array} { r r r r | r } 1 & 0 & 0 & 0 & 3 \\0 & 1 & 0 & 0 & - 10 \\0 & 0 & 1 & 0 & 11 \\0 & 0 & 0 & 1 & - 9\end{array} \right]$ If a system is inconsistent, so indicate.

The matrix is the reduced echelon matrix for a system with variables x 1 , x 2 , x 3 , and x 4 . Find the solution set of the system. $\left[ \begin{array} { r r r r | r } 1 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & - 8 \\0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 & - 9\end{array} \right]$

A box contains $7.40 in nickels, dimes, and quarters. There are 44 coins in all, and the sum of the numbers of nickels and dimes is two less than the number of quarters. How many coins of each kind are there? Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the numbers of nickels, dimes, and quarters, respectively.

Use the appropriate property of determinants from this section to justify the true statement. $( - 4 ) \left| \begin{array} { c c c } 5 & 1 & - 1 \\7 & 5 & 1 \\5 & 1 & 7\end{array} \right| = \left| \begin{array} { c c c } 5 & - 4 & - 1 \\7 & - 20 & 1 \\5 & - 4 & 7\end{array} \right|$ Do not evaluate the determinants. Property number 1: If any row (or column) of a square matrix A contains only zeros, then | A | = 0. Property number 2: If square matrix B is obtained from square matrix A by interchanging two rows (or two columns), then | B | = $-$ | A |. Property number 3: If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k , then | B | = k | A |. Property number 4: If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A , then | B | = | A |. Property number 5: If two rows (or columns) of a square matrix A are identical, then | A | = 0. Enter property number only.

The matrix is in reduced echelon form. $\left. \left[ \begin{array} { l l l } 1 & 0 & 0 \\0 & 0 & 0 \\0 & 1 & 0\end{array} \right] \quad \begin{array} { l } 8 \\0 \\5\end{array} \right]$

Solve the system by the substitution or elimination method. $\left( \begin{array} { l } y = x ^ { 2 } + 3 x - 8 \\y = x ^ { 2 } + 5 x + 10\end{array} \right)$

Evaluate $4 \times 4$ determinant. $\left| \begin{array} { c c c c } 1 & - 2 & 4 & 2 \\2 & - 1 & 0 & 5 \\- 4 & 5 & 0 & - 2 \\- 1 & 1 & 1 & 7\end{array} \right|$ Use the properties of determinants to your advantage. Please enter your answer as a number.

Solve the system by using the elimination-by-addition method. $\left( \begin{array} { l } 7 x &-& y &= &29 \\6 x &+ &3 y& = &21\end{array} \right)$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Change the augmented matrix of the system to reduced echelon form and then indicate the solutions of the system. $\left( \begin{array} { r l l l } x - 5 y + 1 z & = & - 8 \\3 x - 5 y - 17 z & = & - 4\end{array} \right)$

Evaluate $3 \times 3$ determinant. $\left| \begin{array} { c c c } - 5 & - 2 & 1 \\8 & 0 & 7 \\2 & 1 & - 6\end{array} \right|$ Use the properties of determinants to your advantage.

Evaluate $2 \times 2$ determinant. $\left| \begin{array} { c c } \frac { 1 } { 4 } & \frac { 1 } { 6 } \\- 6 & - 24\end{array} \right|$ Please enter your answer as a number.

Solve the homogeneous system. $\left( \begin{array} { l } 5 x+y+z=0 \\6 x+2 y+8 z=0 \\10 x-10 y+z=0\end{array} \right)$

Solve the system. $\left( \begin{array} { r l } 3 x - 2 y + 3 z & = 40 \\5 y - 5 z & = - 50 \\4 z & = 20\end{array} \right)$

A gift store is making a mixture of almonds, pecans, and peanuts, which sell for $3.50 per pound, $5.00 per pound, and $1.00 per pound, respectively. The storekeeper wants to make 15 pounds of the mix to sell at $2.30 per pound. The number of pounds of peanuts is to be three times the number of pounds of pecans. Find the number of pounds of each to be used in the mixture.

Evaluate $2 \times 2$ determinant. $\left| \begin{array} { c c } - 4 & - 6 \\- 3 & - 9\end{array} \right|$ Please enter your answer as a number.

Evaluate $2 \times 2$ determinant. $\left|\begin{array} { c c } \frac { 1 } { 5 } & \frac { 5 } { 14 } \\\frac { 14 } { 25 } & - \frac { 1 } { 14 }\end{array}\right|$ Please enter your answer as a fraction with lowest terms.

Indicate whether the matrix is in reduced echelon form. $\left. \left[ \begin{array} { c c c } 1 & 0 & 0 \\0 & 0 & 0 \\0 & 1 & 0\end{array} \right] \quad \begin{array} { r } 15 \\0\\-14\end{array} \right]$ Enter yes or no .

Michelle can enter a small business as a full partner and receive a salary of $40,000 a year and 10% of the year's profit, or she can be sales manager for a salary of $75,000 plus 5% of the year's profit. What must the year's profit be for her total earnings to be the same whether she is a full partner or a sales manager? Year's profit = $__________

Use determinants and your calculator to solve the following system. $\left( \begin{array} { l } 2.73 x + 2.93 y + 1.57 z = 18.97 \\3.46 x + 3.01 y + 2.32 z = 24.99 \\1.75 x + 3.59 y + 3.29 z = 25.29\end{array} \right)$