6.174 Walk-Through: Finding Eigenvectors and Eigenvalues. In this problem you're going to find the eigenvectors and eigenvalues - (8 i). (a) Write the eigenvalue equation My = Av for this matrix, using the unknowns x and y for the elements of the eigen- vector, and multiply it out to two sep- arate algebra equations. bro of the matrix M = (b) Collect like terms until both equations have the form ax + by = 0. Note that x = y = 0 is a valid solution to these equations. This represents the "trivial" solution that doesn't count as anto eigenvector. ameldo:9.5.8.6 (c) For these two linear equations to have more solutions, they must be linearly dependent: that is, the determinant of their matrix of coefficients must be zero. Write the resulting equation for A. bas = (d) Solve for A. The result should give you two eigenvalues. (e) For each eigenvalue, find the eigenvector: i. Plug the eigenvalue into one of the equations of the form ax + by = 0. Solve to find a relation- ship between x and y. ii. Check that the other equation gives you the same relationship. That tells you that you correctly found an eigen- value, and that any vector (x, y) that satisfies that relationship is an eigen- vector with this eigenvalue. ( that sat- iii. Write an eigenvector () isfies the relationship you found. iv. Confirm that your eigenvalue and eigenvector satisfy the eigenvalue equation for this matrix.