91Ó°ÊÓ

Solve the indicated systems of equations using the inverse of the coefficient matrix. For the following system of equations, solve for \(x^{2}\) and \(y^{2}\) using the matrix methods of this section, and then solve for \(x\) and \(y\). $$\begin{aligned}&x^{2}-y^{2}=8\\\ &x^{2}+y^{2}=10\end{aligned}$$

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

Solve the given problems using Gaussian elimination. Solve the system \(x+2 y=6,2 x+a y=4\) and show that the solution depends on the value of \(a\). What value of \(a\) does the solution show may not be used?

$$\text {Find } B A^{-1}$$$$B=\left[\begin{array}{cc}8 & -2 \\\3 & 4 \end{array}\right] \quad C=\left[\begin{array}{rrr}5 & -1 & 0 \\\2 & -2 & 1 \\\\-3 & 0 & 4 \end{array}\right]$$$$A=\left[\begin{array}{rr}-4 & 1 \\\6 & -2\end{array}\right]$$

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 \(A^{2}=A\)

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

Solve the given systems of equations by determinants. Evaluate by expansion by minors. $$\begin{aligned} &2 x+2 t=0\\\ &3 x+y+z=-1\\\ &2 y-z+3 t=1\\\ &2 z-3 t=1 \end{aligned}$$

Set up systems of equations and solve by Gaussian elimination. Two jets are \(2370 \mathrm{km}\) apart and traveling toward each other, one at \(720 \mathrm{km} / \mathrm{h}\) and the other at \(860 \mathrm{km} / \mathrm{h}\). How far does each travel before they pass?

Find \(B A^{-1}\). In Exercises \(32-34,\) find \(C A^{-1}\) $$B=\left[\begin{array}{ll}8 & -2 \\ 3 & 4\end{array}\right]$$ $$C=\left[\begin{array}{rrr}5 & -1 & 0 \\ 2 & -2 & 1 \\ -3 & 0 & 4\end{array}\right]$$ $$A=\left[\begin{array}{rrr} -3 & 1 & -1 \\ 1 & -4 & -7 \\ 1 & 2 & 5 \end{array}\right]$$

Solve the given systems of equations by determinants. Evaluate by expansion by minors. $$\begin{aligned} &2 x+y+z=4\\\ &2 y-2 z-t=3\\\ &3 y-3 z+2 t=1\\\ &6 x-y+t=0 \end{aligned}$$