Exam 7: Systems of Equations and Matrices

Solve using Cramer's Rule. -The perimeter of a rectangle is 19 meters. If the width is doubled and the length is tripled, the perimeter is 50 meters. Determine the original width and length.

A matrix has an inverse if and only if its determinant does not equal 0. Determine whether the given matrix has an inverse. - $\left[ \begin{array} { l l } 7 & 4 \\6 & 9\end{array} \right]$

Compute AB, if possible. - $A = \left[ \begin{array} { r r } 0 & - 1 \\ 5 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } - 2 & 0 \\ - 1 & 1 \end{array} \right]$

Solve for x. - $\left| \begin{array} { l l l } 4 & x & 1 \\- 4 & 1 & 0 \\5 & 1 & 3\end{array} \right| = 39$

Find the indicated sum or difference, if it is defined. - $\left[ \begin{array} { r } 3 \\- 1 \\- 4\end{array} \right] + \left[ \begin{array} { r } - 5 \\6 \\9\end{array} \right]$

We have encoded a message by assigning the numbers 1 - 26 to the letters a - z of the alphabet, respectively, and assigning 27 to a blank space. We have further encoded it by using an encoding matrix. Decode this message by finding the inverse of the encoding matrix and multiplying it times the coded message. -The encoding matrix i $A = \left[ \begin{array} { r r } 3 & - 2 \\- 4 & 3\end{array} \right] \text { and the encoded message is } 45 , - 55,9 , - 3 , - 33,50 , - 39,61 , - 13,24$

Decide whether or not matrix A and matrix B are inverses. - $\mathrm { A } = \left[ \begin{array} { l l } - 5 & 1 \\- 7 & 1\end{array} \right] \text { and } \mathrm { B } = \left[ \begin{array} { l l } 0.5 & - 0.5 \\3.5 & - 2.5\end{array} \right]$

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { c } 4 x + 7 y - z = 61 \\x - 3 y - 8 z = - 21 \\- 9 x + y + z = - 38\end{array} \right.$

The matrix associated with the solution to a system of linear equations in x, y, and z is given. Write the solution to the system, if it exists. - $\left[ \begin{array} { l l l | r } 1 & 0 & 0 & 3 \\0 & 1 & 0 & - 14 \\0 & 0 & 1 & 1\end{array} \right]$

An open topped box with a square base will have volume of 9464 cubic inches and uses 2132 square inches of material. What are the dimensions of the box?

Find the indicated matrix. -Let $A = \left[ \begin{array} { l l } - 1 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } 1 & 0 \end{array} \right]$ . Find $3 A + 4 B$ .

Solve using Cramer's rule, if possible. If it is not possible, state whether the system is inconsistent or has infinitely many solutions. - $\left\{ \begin{array} { l } x + y = 5 \\3 x + 3 y = 15\end{array} \right.$

Compute AB, if possible. - $A = \left[ \begin{array} { r r } - 1 & 3 \\ 2 & 2 \end{array} \right]$ and $B = \left[ \begin{array} { l l } - 2 & 0 \\ - 1 & 2 \end{array} \right]$

A trust account manager has $300,000 to be invested in three different accounts. The accounts pay 5%, 6%, and 8%, and the goal is to earn $18,000. Assuming that x dollars are invested at 5%, y dollars are invested at 6%, and Z dollars are invested at 8%, find x in terms of z.

A matrix has an inverse if and only if its determinant does not equal 0. Determine whether the given matrix has an inverse. - $\left[ \begin{array} { r r } 7 & - 4 \\- 7 & 5\end{array} \right]$

A theater has 975 seats divided into orchestra, main, and balcony. The orchestra seats cost $85, the main seats cost $55, and the balcony seats cost $45. If all the seats are sold, the revenue is $54,125. If all the orchestra and Balcony seats are sold and 3/5 of the main seats are sold, the revenue is $42,575. How many balcony seats does The theater have?