Answer Happy

4. Let A = 3 1 2 2 2 2 2 1 3 You are given that A has eigenvalues 1 and 6, with eigenspaces -1 --4-7--0)emm*- {-() ---} {(( 7 : :' • : E a +B aBER and E6 8 :8 ER (a) Is A invertible? (You can do this without writing anything.) (b) Is A diagonalizable? Explain how you know. (c) Is A orthogonally diagonalizable? Explain how you know. (d) How many different such matrices P and D can you find satisfying P-1AP = D? Explain. -1 -1 1 1 0 0 (e) For the rest of the question, use P = 2 0 1 and D= 0 1 0 1 1 0 0 6 Do you agree that would give P-AP = D? The only reason I am specify- ing this now, is that I want to save you some time in the calculations later -1 -1 1 on: at some stage, you may want to use the fact that 2 0 1 0 1 1 2 -1 1 -2 -1 3 5 2 1 2 0 -1 -1 Use your results above to solve the system of differential equations a't 3.x1 + x2 + 2x3 x'a 2x1 + 2x2 + 2x3 2x1 + x2 + 3x3 = (f) Find the particular solution to the system above which also satisfies the con- ditions 21(0) = 2, 27(0) = 1, 23(0) = -10. (g) Modify your working above to solve the system 21 3.x1 + x2 + 2x3 + 5 x's 2.C1 + 2x2 + 2.23 x's = 2x1 + x2 + 3x3 If you would like more practice on diagonalization and using it to solve systems of differential equations, here are some more questions for you to do: