Question

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1. Consider the matrix A = [ 24 31. a) (7 pnts) Find the characteristic polynomial of A. b) (7 pnts) Compute the matrix B = A- 2A +812. c) (6 pnts) Can you describe how to find the inverse of A using characteristic equation?

Answer 1

we have matix Az 2 4 2 0 as the chasactuitić folynomial of Acil bed) det (A-dI) p (d) = 2-4 -4 2 0-2 pld!= (2-d)(ord) + 8 pld) = a(d-2) to ped) = x-2xt8 +8 b) compute the matrix B = A2 - 2A + 8 I. 2 A², A: Az 2 2 2 o 14 O A2 4 -8 - Then we get ។ 2 2 1 B- -4 -8 -2 +0 He -4 0 -4 4 B - -4 -4 + 8 0 -0-8 8 0 0 8 0 8 oo 0

بین c) we have the characteristie fequation of A p(d)=0 ² - 2 +8=0 By layley Hamilton theorum, matrix Asatújy ite charastuise's equation weget A² - 2 At & I2 = 0 Multiply by A! both sides, we get A1 A² - 12 AA + 8 A'I = 0 A.–21+ 8A4 = 0 (A 'Is = 8) TA'A-12) BA' - A +2 Iz Al= Il-a + 2 I.) Alauf (A - 21) This is the equation of Al. 2) as we have y - y =o. the charactuistic equation i m²-1=0 m²=1 m = 1 m = -1 (root) given by m = 1 thep, the general solution is yota sema tent y (2) geet het

b) we know that the solution will be complex only when one characteristic Hequation has no real soits: So, you complex roote, we can change the above differential equation to affuential equation y + y =o c) Now, we have y'-y = 342x In part a), we chare complementary solution of thei system Now by using variation of Parameters to find une particular sondion of system: from part a) the we hare ly, (t) = ex ₂ (t) = ex of these ly, Y' (t) The winskian two functionis 4,(7) et et Y(t) et - et ezel - peek =--= -2 the particular solution is eren, 2 gt att dy y g (t) at - Up Ot) = - Y, + y 2014 W

سواعد علمدم و پله (تعدي (لع (م د) (*علل متعار d - 3 er fer de 3 ef et de حه 3م (2) عدد (1) م و = 2م *م - 28م = The general solution u (1) مو + ( 2) * = 5 لا كم + cente? + lying

Answer 2

b)

B=A²-2A+8I₂

B=A.A-2A+8[1      0

                           0        1 ]

Paul all these vale in this     Question 

Answer 3

b)

B=A²-2A+8I₂

B=A.A-2A+8[1      0

                           0        1 ]

Paul all these vale in this     Question