Exam 15: Systems of Equations: Matrices and Determinants

Part of $5,000 is invested at 9%, another part at 10%, and the remainder at 11% yearly interest. The total yearly income from the three investments is $505. The sum of the amounts invested at 9% and 10% equals the amount invested at 11%. How much is invested at each rate? Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the amounts invested at 9%, 10%, and 11%, respectively. Please do not include units in your answer. $__________, $__________, $__________

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

Indicate whether the matrix is in reduced echelon form. $\left. \left[ \begin{array} { c c c } 1 & 1 & 0 \\0 & 1 & 3 \\0 & 0 & 1\end{array} \right] \quad \begin{array} { c } 8 \\8 \\- 2\end{array} \right]$ Enter yes or no .

An inconsistent system has exactly one solution.

Use a matrix approach to solve the system. $\left( \begin{array} { c } x - 3 y - z = 3 \\5 x + y - 6 z = 5 \\- 4 x + 5 y + 5 z = 11\end{array} \right)$

Evaluate $4 \times 4$ determinant. $\left| \begin{array} { c c c c } 1 & 2 & 6 & 8 \\- 7 & 3 & 0 & 9 \\- 3 & 6 & 2 & 8 \\2 & 1 & 5 & 3\end{array} \right|$ Use the properties of determinants to your advantage.

Solve the homogeneous system. $\left( \begin{array} { r } 2 x - y + 2 z = 0 \\x + 7 y + z = 0 \\x - 8 y + z = 0\end{array} \right)$ If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent .

Evaluate $4 \times 4$ determinant. $\left| \begin{array} { c c c c } 3 & - 1 & 2 & 3 \\1 & 0 & 2 & 1 \\2 & 3 & 0 & 1 \\6 & 2 & 4 & - 6\end{array} \right|$ Use the properties of determinants to your advantage. Please enter your answer as a number.

The determinant of any matrix with real number entries is a real number.

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l } 16 x ^ { 2 } + 2 y ^ { 2 } & = 70 \\y ^ { 2 } - 16 x ^ { 2 } & = 38\end{array} \right)$

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

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

Evaluate $3 \times 3$ determinant. $\left| \begin{array} { c c c } 5 & - 7 & - 4 \\9 & - 4 & 3 \\3 & 0 & 0\end{array} \right|$ Use the properties of determinants to your advantage. Please enter your answer as a number.

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { c c } y & =& 4 x \\0.71 x & +& 0.06 y &= &475\end{array} \right)$ 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 ).

Solve the system by the substitution or elimination method. $\left( \begin{array} { r l l l } 5 y ^ { 2 } - 8 x ^ { 2 } & = & 31 \\8 x ^ { 2 } - 2 y ^ { 2 } & = & 2\end{array} \right)$

The matrix is in reduced echelon form. $\left. \left[ \begin{array} { c c c c } 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\end{array} \right] \quad \begin{array} { r } 1 \\- 7 \\1\\-8\end{array} \right]$

The solution set for the system $\left( \begin{array} { c } x ^ { 2 } + y ^ { 2 } = 9 \\x ^ { 2 } + y ^ { 2 } = 17\end{array} \right)$ is the null set.

Solve the system. $\left( \begin{array} { r l } 3 x - 2 y + 3 z & = 23 \\3 y - 5 z & = - 18 \\2 z & = 6\end{array} \right)$

Use Cramer s rule to find the solution set for the system. $\left( \begin{array} { c c c c } x - 5 y & = & - 16 \\5 x + 3 y & = & 32\end{array} \right)$ 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 the system by using the elimination-by- addition method. $\left( \begin{array} { c c c c } x &-& 9 y & = & 62 \\2 x& + &8 y & = & - 58\end{array} \right)$ 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 ).