Exam 15: Systems of Equations: Matrices and Determinants

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { c } \frac { 4 x } { 5 }& - &\frac { 3 y } { 2 }& = &\frac { 1 } { 5 } \\- 2 x &+& y& =& - 3\end{array} \right)$ 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 ), enter x and y as fractions.

Evaluate $3 \times 3$ determinant. $\left| \begin{array} { c c c } 4 & - 1 & 5 \\0 & 5 & 1 \\1 & - 4 & - 1\end{array} \right|$ Use the properties of determinants to your advantage.

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. $\left[ \begin{array} { r r r r | r } 1 & 0 & 0 & 10 & - 8 \\0 & 1 & 0 & 0 & 5 \\0 & 0 & 1 & 6 & 3 \\0 & 0 & 0 & 0 & 0\end{array} \right]$

A small company makes three different types of bird houses. Each type requires the services of three different departments, as indicated by the following table. Type A Type B Type C Cutting department 0.1 hour 0.1 hour 0.2 hour Finishing department 0.5 hour 0.5 hour 0.3 hour Assembly department 0.3 hour 0.1 hour 0.2 hour The cutting, finishing, and assembly departments have available a maximum of 27, 86, and 37 work-hours per week, respectively. How many bird houses of each type should be made per week so that the company is operating at full capacity? Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the numbers of houses of type A, B, and C, respectively.

Solve the system. $\left(\begin{array}{rlll}4 x&+&2 y&-&z & = & -4 \\4 x&-&4 y&+&4 z & =&40 \\5 x&+&y&-&5 z & = & -14\end{array}\right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { l } y = - x ^ { 2 } + 3 \\y = x ^ { 2 } + 1\end{array} \right)$ Express your answer as an ordered pair. If there is more than one solution, separate your answers with a comma.

Use Cramer s rule to find the solution set for the system. $\left( \begin{array} { r r r r } - \frac { 4 } { 5 } x &+& \frac { 1 } { 2 } y & =& - 10 \\\frac { 1 } { 5 } x &-& \frac { 3 } { 2 } y & = & - 3\end{array} \right)$ 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 in the form ( x , y ).

Solve each system of equations by the substitution or elimination method. Match each system of equations with the corresponding answer. - $( 4 , - 24 )$

Use the appropriate property of determinants from this section to justify the true statement. $\left| \begin{array} { c c c } 1 & 4 & 6 \\- 2 & 7 & 8 \\- 4 & - 1 & 2\end{array} \right| = \left| \begin{array} { c c c } 1 & 4 & 6 \\- 2 & 7 & 8 \\0 & 15 & 26\end{array} \right|$ 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.

Solve the system by the substitution or elimination method. $\left( \begin{array} { l } x + y = - 5 \\x ^ { 2 } + 2 y ^ { 2 } - 7 y - 151 = 0\end{array} \right)$ Express your answer as an ordered pair. If there is more than one solution, separate your answers with a comma.

Solve the system. $\left(\begin{array}{rlll}2 x&+&5 y&-&3 z & =&-13 \\5 x&-&2 y&+&4 z & =&34 \\&&4 y&+&z & =&-10\end{array}\right)$

Evaluate $3 \times 3$ determinant. $\left| \begin{array} { c c c } - 5 & - 4 & 3 \\8 & 0 & 7 \\4 & 3 & - 6\end{array} \right|$ Use the properties of determinants to your advantage. Please enter your answer as a number.

Part of $10,000 is invested at 9%, another part at 11%, and the remainder at 12% yearly interest. The total yearly income from the three investments is $1,100. The sum of the amounts invested at 9% and 11% equals the amount invested at 12%. How much is invested at each rate?

A determinant can be calculated for any square matrix.

Solve the system. $\left( \begin{array} { r l l } 2 x&-&4 y&+&2 z&=&0 \\4 x&+&2 y&-&6 z&=&20 \\5 x&-&3 y&+&6 &z=&-10\end{array} \right)$

Use Cramer s rule to find the solution set for the system. $\left( \begin{array} { l } \frac { 1 } { 2 } x + \frac { 2 } { 3 } y = 14 \\\frac { 1 } { 4 } x - \frac { 1 } { 3 } y = 1\end{array} \right)$ 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 in the form ( x , y ).

Evaluate $2 \times 2$ determinant. $\left| \begin{array} { r r } 4 & - 6 \\7 & - 4\end{array} \right|$ Please enter your answer as a number.

The measure of the largest angle of a triangle is twice the measure of the smallest angle. The sum of the smallest angle and the largest angle is twice the other angle. Find the measure of each angle.

Solve the homogeneous system. $\left( \begin{array} { r } 2 x - y + 2 z = 0 \\x + 6 y + z = 0 \\x - 7 y + z = 0\end{array} \right)$

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l r } x ^ { 2 } + y ^ { 2 } & = 61 \\x + y & = - 1\end{array} \right)$ Express your answer as an ordered pair. If there is more than one solution, separate your answers with a comma.