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10 sine (adja = Adjoint of A del A - Determinant of A given: A=11 3 A - adja det A 4 27 - 0 - A = Then, det = 1(-2) -4 (6) +2 (3+1) = -2 -2448 = -18. adj A - (cofactor) = -2 -6 -21 1-6 0 6 4 +3 -13 [ua 113 19 7 113 o 1/3 [29 -16 13/18) To compute : Ax=b where A is as defined above and b= (1,2,3) equivalent to solve x = Alb. (say), x= (%, , 22, 3) [110 113 va 17 ( 1% 7. 13 0 132 - 2/3 3 1 -29 -116 13/10) (²) (29/8) C which is required. 2 (a) v is an eigenvector of B ifth 7 à st. Bu, = 1, U, - (A) 0,70) first, let us calculate the eigenvalues of B which we know is given by : det (A1-B) = 0 (Roots of this characteristic esh) Those are & d=-1, de 3, dz= 5. using statement (A), we can easily calculate, Eigenvector for = 1 : Eigenvector for 1-3 : Eigen vector for 1 5 : So, ginen v, is an eigen vector correspondiup to 1,5 & given 2 1 " (6) V3 = [0,-4,-4] = Uv, - u V2 Scanned with = 4[1, 1,0] -4 [1, 2,1 ] CamScanner
Hence, Blot = 810(4V, -4₂) - UBIOU, UBIO U2 = 4(B) (BV) - B9 Bv2) - 4 CB (₂ V;) - B9 (1,02)) V, corresponds toda) & V₂ " " di) - 4 (B950, - B9 (12) = u(510 v, -1-111002) Hence, BiO V₂ = 41510 v, -V2) 4 (slovi - 12) which complete pout (2). (3rd is done after it) 6 et D= doon 020 oo 13 clearly, the eigenvalues of this diagonal matrix are d., 12, 13. for eigenvectors, first fordi det v- (V, V2, V3T is eigenvector DV=div do ou NAV) 1, V2 - Lo o d [v diva a ijine divi = divi A2/2 = 1102, & 3 3 = 1103 = since d e , de are distinct real nos od 0 V, Hence, (1,0,0) is eigenvector corr. to di (0,1,0) de 0,0,0T , de Hence, true for a general diagonal matrix. 5 This part is by the application of Gershgorin Theorem which states that if A= (aij) and Pia &laij) then it is an eigenvalue of a then, min (20ij - Pi) slll a max Pi . a hell with which is every now sum for Elx4), is an Camsleanielu Oy E and correspondine eigenvector is LOT C
12 0 (trc = trace of c) -10 Characteristic polyn of the matrix c = det (AI-C). = x (trc) & + det o 1+1 eigenvalues of c are roots of 1²+1=0 ie 12 = -1 a d= + - 1 = fi - C has complex eigenvalues. for eigenvector corresponding to ti! cx = in a lon - Lix] a) x2 = Taking either of x, or 2 w real ag it ai is real then . 2 is complex & if He is real then - 2 is complex ing=in, & eigenvectos corresponding to i au of the form on * Similarly , ciganuectors correspondiup to oi au of Hu form Hence, for both the eigenvalues, c has no non-zero eigenvector with real entries. And * '**' are the complex -values eigenvector of a for 2, ¢. 1 Di This completes (3). Scanned with CamScanner