3.4E: Exercises - Solving Systems with Inverses
Find the inverse of each matrix (if possible).
1. \[A=\left [ \begin{array}{rr} 2 & 1 \\ -1 & 3 \end{array} \right ]\nonumber\]
2. \[A=\left [ \begin{array}{rr} 0 & 1 \\ 5 & 3 \end{array} \right ]\nonumber\]
3. \[A=\left [ \begin{array}{rr} 2 & 1 \\ 4 & 2 \end{array} \right ]\nonumber\]
4. \[A=\left [ \begin{array}{rrr} 1 & 0 & 3 \\ 2 & 3 & 4 \\ 1 & 0 & 2 \end{array} \right ]\nonumber\]
Use inverses to solve for all variables.
5.
\(\begin{bmatrix}-3&1 \nonumber \\[4pt] 4&-2\end{bmatrix}\begin{bmatrix}x \nonumber \\[4pt] y\end{bmatrix}=\begin{bmatrix}-7 \nonumber \\[4pt] 12\end{bmatrix}\)
6. \[\begin{align*} 3x+y&= 2\\ 10x+7y&= -8 \end{align*}\nonumber\]
7. \[\begin{align*} x+6y+3z&=4\\ 2x+y+2z&=3\\3x-2y+z&=0 \end{align*}\nonumber\]
Write a system of equations to represent each scenario. Then use inverses to solve for the desired quantity.
8) A fast-food restaurant has a cost of production \(C(x)=11x+120\) and a revenue function \(R(x)=5x\). When does the company start to turn a profit?
9) A musician charges \(C(x)=64x+20,000\), where \(x\) is the total number of attendees at the concert. The venue charges \(\$80\) per ticket. After how many people buy tickets does the venue break even, and what is the value of the total tickets sold at that point?
10) If an investor invests \(\$23,000\) into two bonds, one that pays \(4\%\) in simple interest, and the other paying \(2\%\) simple interest, and the investor earns \(\$710.00\) annual interest, how much was invested in each account?