final extra-Solution - MATH 20F LINEAR ALGEBRA(WINTER 2016 SOME EXTRA EXERCISES ON THE NEW MATERIALS First some warnings before we start Eigenvectors

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MATH 20F: LINEAR ALGEBRA (WINTER 2016) SOME EXTRA EXERCISES ON THE NEW MATERIALS First, some warnings before we start: Eigenvectors cannot be zero vector. I have an obsessive hatred for zero eigenvector so if I see them in any problem that I grade, I will take off all points for that part, no partial credit, no exception! Eigenvalues can still be zero. If A is a matrix with a zero eigenvalue, then det ( A ) = 0 and A is singular/non-invertible. When finding the eigenvalue through the determinant det ( A - λ I ) , do NOT use row re- duction. Use cofactor expansion instead. Just like with the columns space, row operations WILL change the eigenvalues of a Matrix. Problem 1. Let v 1 = 1 1 1 , v 2 = - 2 1 1 , and v 3 = 0 1 - 1 . Show that B = { v 1 , v 2 , v 3 } is an orthogonal set in R 3 and find the distance between v 2 and v 3 . Solution. Problem 2.LetA=-2425.a. Find the characteristic polynomial ofAb. Show that 6 and-3 are eigenvalues ofAand find their corresponding eigenvectors. Solution. 1