91Ó°ÊÓ

Set up systems of equations and solve by Gaussian elimination. The voltage across an electric resistor equals the current (in \(\mathbf{A}\) ) times the resistance (in \(\Omega\) ). If a current of 3.00 A passes through each of two resistors, the sum of the voltages is \(10.5 \mathrm{V}\). If \(2.00 \mathrm{A}\) passes through the first resistor and 4.00 A passes through the second resistor, the sum of the voltages is \(13.0 \mathrm{V}\). Find the resistances.

Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix \(A, A^{2}=A \times A.\) $$A=\left[\begin{array}{rrr}2 & -3 & -5 \\\\-1 & 4 & 5 \\\1 & -3 & -4\end{array}\right] B=\left[\begin{array}{rrr}1 & -2 & -6 \\\\-3 & 2 & 9 \\\2 & 0 & -3\end{array}\right] C=\left[\begin{array}{rrr}1 & -3 & -4 \\\\-1 & 3 & 4 \\\1 & -3 & -4\end{array}\right]$$ Show that \(C^{2}=O\)

Use the matrices A, B, C and D for Exercises and find the indicated matrices using a calculator. $$\begin{aligned} &A=\left[\begin{array}{ll} 6 & -3 \\ 4 & -5 \end{array}\right] \quad B=\left[\begin{array}{rr} 3 & 12 \\ -9 & -6 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} -1 & 4 & -7 \\ 2 & -6 & 11 \end{array}\right] \quad D=\left[\begin{array}{rrr} 7 & 9 & -6 \\ -4 & 0 & 8 \end{array}\right] \end{aligned}$$ $$-4 C-3 D$$

Solve the indicated systems of equations using the inverse of the coefficient matrix. In solving the system of equations \(3 x-4 y=5,8 y-6 x=7\), what conclusion can be drawn?

Solve the given systems of equations by determinants. Evaluate by expansion by minors. $$\begin{aligned} &x+2 y-z=6\\\ &y-2 z-3 t=-5\\\ &3 x-2 y+t=2\\\ &2 x+y+z-t=0 \end{aligned}$$

Perform the indicated matrix multiplications on a calculator, using the following matrices. For matrix \(A, A^{2}=A \times A.\) $$A=\left[\begin{array}{rrr}2 & -3 & -5 \\\\-1 & 4 & 5 \\\1 & -3 & -4\end{array}\right] B=\left[\begin{array}{rrr}1 & -2 & -6 \\\\-3 & 2 & 9 \\\2 & 0 & -3\end{array}\right] C=\left[\begin{array}{rrr}1 & -3 & -4 \\\\-1 & 3 & 4 \\\1 & -3 & -4\end{array}\right]$$ Show that \(B^{3}=B\)

Use the matrices A, B, C and D for Exercises and find the indicated matrices using a calculator. $$\begin{aligned} &A=\left[\begin{array}{ll} 6 & -3 \\ 4 & -5 \end{array}\right] \quad B=\left[\begin{array}{rr} 3 & 12 \\ -9 & -6 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} -1 & 4 & -7 \\ 2 & -6 & 11 \end{array}\right] \quad D=\left[\begin{array}{rrr} 7 & 9 & -6 \\ -4 & 0 & 8 \end{array}\right] \end{aligned}$$ $$-6 B-4 A$$

Set up systems of equations and solve by Gaussian elimination. Three machines together produce 650 parts each hour. Twice the production of the second machine is 10 parts/h more than the sum of the production of the other two machines. If the first operates for \(3.00 \mathrm{h}\) and the others operate for \(2.00 \mathrm{h}, 1550\) parts are produced. Find the production rate of each machine.

Solve the indicated systems of equations using the inverse of the coefficient matrix. Forces \(\mathbf{A}\) and \(\mathbf{B}\) hold up a beam that weighs \(2540 \mathrm{N},\) as shown in Fig. 16.16. The equations used to find the forces are $$\begin{array}{l}A \sin 47.2^{\circ}+B \sin 64.4^{\circ}=2540 \\\A \cos 47.2^{\circ}-B \cos 64.4^{\circ}=0\end{array}$$ Find the magnitude of each force.

Find \(B A^{-1}\). In Exercises \(32-34,\) find \(C A^{-1}\) $$B=\left[\begin{array}{ll}8 & -2 \\ 3 & 4\end{array}\right]$$ $$C=\left[\begin{array}{rrr}5 & -1 & 0 \\ 2 & -2 & 1 \\ -3 & 0 & 4\end{array}\right]$$ $$A=\left[\begin{array}{rrr} 1 & -1 & 1 \\ 0 & -2 & 1 \\ -2 & -3 & 0 \end{array}\right]$$