2 11 Diagonalize the matrix M = [ TAKE-HOME ASSIGNMENT 2 MAT223 - WINTER 2024 Please make sure to read the Assignment Instructions posted on Quercus to find out how and when to submit your assignment, what is expected of you, etc. This question is optional. ?: x:], and use this to determine a formula for M2 (You may leave expressions involving large powers, like 2202, unsimplified.) Determine an expression for MF (for a fixed k > 0) using your answer to the previous part. (This means that you should write ME as a matrix with 'K' only k k (=1 100" =) appearing on the inside of the matrix, like 1/3 Use your previous answer to determine Mixask forx = L . You do not need to prove that your answer is correct. (Note: you can take this question to mean \"what matrix does M*x seem to be getting closer to as k gets very large? It might help to write it down for some specific values of k") 2 Suppose that L is the line through point {1,2,3) with direction vector |3]. Lo And suppose that L; is the line given by the intersection of the two planes x+yz=land2xy+z=2 1 Show that Ly and Ls do not intersect. Explain why Ly and Lz are not parallel. i Determine the shortest distance between any two points, Py and Pa, where Py lies on Ly and I lies on Ly. (We could just call this "the shortest distance between Ly and L,".) You do not have to find the points Py and Py. Hint: there is an Example in the textbook that covers a similar problem. I In this question we introduce a new definition. Definition: We say that an n x n matrix A is sharp if A is diagonalizable, and A = I. 1 0 0 5.1 Optional: Show that A = |0 1 0 | is sharp. 00 -1 Optional: Find all diagonal 2 x 2 sharp matrices. Because a sharp matrix is diagonalizable, we can write A = PDP-!, where D is diagonal and P is invertible. Prove from this that for such a matrix D, DA = I (and so [ is also sharp). Prove that the only possible (real number) eigenvalues for a sharp matrix are +1. Explain why any sharp matrix must be invertible. As part of your answer (or additionally), determine a formula for A7Vin terms of A (like 2A + I'or "A + A%, etc.) Construct a non-diagonal 2 x 2 sharp matrix. To do this, you can take P to have two non-parallel vectors vy, vy R? which are not parallel to either e; or ;. Then choose an appropriate [? {(you'll have to think about what it should be) and then set A tobe PDP~! {where P and D) are the matrices you created.) Explain in your own words how you know that the matrix A you created is indeed a non-diagonal, sharp matrix. Determine if the statements below are True or False. If it's True, explain why. If it's False explain why not, or give an example demonstrating why it's false with an explanation. A correct choice of \"True\" or \"False\" with no explanation will not receive any credit. | True or False: If A is a 3 3 matrix with characteristic polynomial c4(x) = 12, then A is the zero matrix. * True or False: If Py and P are distinct planes in R? so that for every line Ly contained in P there is a line Ly contained in P; so that Ly and Ly are parallel, then Py and P must be ]J.atral!.el.l