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Chapter 3: Problem 7
Find the determinant of the matrix. $$\left[\begin{array}{rr} -7 & 6 \\ \frac{1}{2} & 3 \end{array}\right]$$
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Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{rr} -2 & 4 \\ 2 & 5 \end{array}\right]$$
Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{aligned} 3 x_{1}-2 x_{2}+x_{3} &=-29 \\ -4 x_{1}+x_{2}-3 x_{3} &=37 \\ x_{1}-5 x_{2}+x_{3} &=-24 \end{aligned}$$
Use a graphing utility or a computer software program with matrix capabilities and Cramer's Rule to solve for \(x_{1}\) if possible. $$\begin{array}{l} 4 x_{1}-x_{2}+x_{3}=-5 \\ 2 x_{1}+2 x_{2}+3 x_{3}=10 \\ 5 x_{1}-2 x_{2}+6 x_{3}=1 \end{array}$$
Find (a) the characteristic equation, (b) the eigenvalues, and (c) the corresponding eigenvectors of the matrix. $$\left[\begin{array}{ll} 3 & -1 \\ 5 & -3 \end{array}\right]$$
Determine whether the points are coplanar $$(1,2,7),(-3,6,6),(4,4,2),(3,3,4)$$
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