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To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 1 and 2, use the power method as illustrated in Example 3 to find the dominant eigenvalue and a corresponding dominant eigenvector of the given matrix. \(\left(\begin{array}{ll} 1 & 1 \\ 2 & 0 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 1 and 2, use the power method as illustrated in Example 3 to find the dominant eigenvalue and a corresponding dominant eigenvector of the given matrix. \(\left(\begin{array}{rr} -7 & 2 \\ 8 & -1 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 3–6, use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. \(\left(\begin{array}{rr} 2 & 4 \\ 3 & 13 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 3–6, use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. \(\left(\begin{array}{ll} -1 & 2 \\ -2 & 7 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 3–6, use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. \(\left(\begin{array}{lll} 5 & 4 & 2 \\ 4 & 5 & 2 \\ 2 & 2 & 2 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 3–6, use the power method with scaling to find the dominant eigenvalue and a corresponding eigenvector of the given matrix. \(\left(\begin{array}{lll} 3 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 7–10, use the method of deflation to find the eigenvalues of the given matrix. \(\left(\begin{array}{ll} 3 & 2 \\ 2 & 6 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 7–10, use the method of deflation to find the eigenvalues of the given matrix. \(\left(\begin{array}{ll} 1 & 3 \\ 3 & 9 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 7–10, use the method of deflation to find the eigenvalues of the given matrix. \(\left(\begin{array}{rrr} 3 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 3 \end{array}\right)\)

To the Instructor/Student: A calculator with matrix capabilities or a CAS would be useful in the following problems. Each matrix in Problems 1–10 has a dominant eigenvalue. In Problems 7–10, use the method of deflation to find the eigenvalues of the given matrix. \(\left(\begin{array}{rrr} 0 & 0 & -4 \\ 0 & -4 & 0 \\ -4 & 0 & 15 \end{array}\right)\)

In Problems 11 and 12, use the inverse power method to find the eigenvalue of least magnitude for the given matrix. \(\left(\begin{array}{ll} 1 & 1 \\ 3 & 4 \end{array}\right)\)

In Problems 11 and 12, use the inverse power method to find the eigenvalue of least magnitude for the given matrix. \(\left(\begin{array}{rr} -0.2 & 0.3 \\ 0.4 & -0.1 \end{array}\right)\)

In Example 4 of Section 3.9 we saw that the deflection curve of a thin column under an applied load P was defined by the boundary-value problem \(E I \frac{d^{2} y}{d x^{2}}+P y=0, \quad y(0)=0, \quad y(L)=0\). In this problem we show how to apply matrix techniques to compute the smallest critical load. Let the interval [0, L] be divided into n subintervals of length h = L/n, and let \(x_i = ih\), i = 0, 1, . . . , n. For small values of h it follows from (6) of Section 6.5 that \(\frac{d^{2} y}{d x^{2}} \approx \frac{y_{i+1}-2 y_{i}+y_{i-1}}{h^{2}}\), where \(y_i = y(x_i)\). (a) Show that the differential equation can be replaced by the difference equation \(E I\left(y_{i+1}-2 y_{i}+y_{i-1}\right)+P h^{2} y_{i}=0, i=1,2, \ldots, n-1\). (b) Show that for n = 4 the difference equation in part (a) yields the system of linear equations \(\left(\begin{array}{rrr} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{array}\right)\left(\begin{array}{l} y_{1} \\ y_{2} \\ y_{3} \end{array}\right)=\frac{P L^{2}}{16 E I}\left(\begin{array}{l} y_{1} \\ y_{2} \\ y_{3} \end{array}\right)\). Note that this system has the form of the eigenvalue problem \(AY = \lambdaY\), where \(\lambda =PL^2/16EI\). (c) Find \(A^{-1}\). (d) Use the inverse power method to find, to two decimal places, the eigenvalue of A of least magnitude. (e) Use the result of part (d) to compute the approximate smallest critical load. Compare your answer with that given in Section 3.9.

Suppose the column in Problem 13 is tapered so that the moment of inertia of a cross-section I varies linearly from \(I(0) = I_0 = 0.002\ \text{to}\ I(L) = I_L = 0.001\). (a) Use the difference equation in part (a) of Problem 13 with n = 4 to set up a system of equations analogous to that given in part (b). (b) Proceed as in Problem 13 to find an approximation to the smallest critical load.

In Section 8.9 we saw how to compute a power \(A^m\) for an \(n\ \times\ n\) matrix A. Consult the documentation for the CAS you have on hand for the command to compute the power \(A^m\). (In Mathematica the command is MatrixPower[A, m].) The matrix \(\mathbf{A}=\left(\begin{array}{rrr} 5 & -2 & 0 \\ -2 & 3 & -1 \\ 0 & -1 & 1 \end{array}\right)\) possesses a dominant eigenvalue. (a) Use a CAS to compute \(A^{10}\). (b) Now use (2), \(X_m = A^mX_0\), with m = 10 and \(\mathbf{X}_{0}=\left(\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right)\), to compute \(X_{10}\). In the same manner compute \(X_{12}\). Then proceed as in (9) to find the approximate dominant eigenvector K. (c) If K is an eigenvector of A, then \(AK = \lambda K\). Use this definition and the result in part (b) to find the dominant eigenvalue.