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clasSMate, clasSMate, Date, Page, Date, Pege, * Spectoum of A-, The set of all eigen volues of, Again do the same procedure, matrix A is called spectrum of A., -1, -3, ---, 4, -4, 101gives again D=-1, And D-40+4%3D0and so on., -4, * Chonacteristic Vector or figen vectomsi-, Let X be the solution vectorof, CA-AI) X =0 corresponding to ch", roots (Ai), then x; is called ch, vectors o Gigen vectors or latent, vector associated oith the root Ai., * Charocteristica polynomial of A -, D(4) = |A-XI, is known, os charractesistic polynomial of A., * Notes -, D expansion of determinant is woitten, by using shortut method as follows:, let, *characteristic Equation of matrix A-, D(y) = LA-XI|=0, 90A+ 9,+--- +an=0 is known, as charactegistic eg of matrix A, The degree of characteristic eg^ of, matrix A is equal to the order Of, that matrix., A =a 912, 92192219x2, then IA-AI =0, ie X-SiA - IA1=0, Where., S, = Sum of principle elements., =Q,922, -2 921, * Chamcteristic Roots m Gigen Values, Let, D(y) = LA-XI, Roots ome (A-Xi) CA-12--(A-An), where A=,22,- Xn are called, charocteristics Tootsare, Or proper values or latent values., 921 a22 923, 933 3x3, A=, 932, then I A-AI| =0, eigen valu, klhere S,= au +922 + 933

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UASSMATE, claSSMate, Date, Date, Poge, Page, 4, 2, - 2, Now. IAI=, -5, 2., Using standard formula:-, -2, 4, Si = Sum of the poincipal, diagonal elements ke, [an+922+983], S2= sum af "minors of the, principol diagonal elements, Where, = 4, 2-2-5 2+(-2)-5 3, 41, -2 I, -2 4, =4(3x1)-(4x2)]- 2[(-5x)- (-2 x2)], +(+2)[(-5x4)-(-2 x 3) ], 4., - 2, = 4[3-8]-2(-5+4]+(-2)[-20+6], = 4(-5)-2(-1)-2(-14), =- 20+2+28, - 5, 2., 6, = SUmof, -2, diagonat etements, IA) 10, Substitute values of S,,S2f1AI, in eg @, = 4+3+1, S, =8, it gives, S2 = [minor of4] +[Minor of 3]+, +[Minor of 1], %3D, Which is required eq"., Now, by using synthetic Divis ion,, 3 2, 4 - 2, 土, %3D, 4, 4 1, -2, -5 3, 17, -10, 1., -8, (sum of minors of diagonal e lemei, 7, 10, -7, 10, +(4x3)-(-5)x2)], 2, (3-8) +(4-4) +(12+10)], -21-5) +10 = 0, (X-2)-5(-2) =0, =-5+0+22, S=17