Question

Construct a non-triangular 3x3 matrix, which has three distinguish eigenvalues. Find corresponding eigenvectors of this matrix.

Answer 1

Let the non- triangular 3x3 matrix ist 2 A = - 2 -4 - 2 4 2 5 2 -1-2 -4 2 | A-11 12 - 2 2 4 2. 5-) IA-XI) - 202 + (1-1) (5-1) (-1-2)- we have to solve the above characteristic Polynomial to find the roots. 7) 201+ (1-1)(5-1) (-1-2) = 5) -33+4x2+372-90=0 13-442- 27 ) + 90 = 0 put x = 3 27-419) - 27X3 +90 = 0 One root So, l=3 is the Nowy By Synthetic division methoda -4 - 27 go -90 - 3 3 | 0 - 30 - 1 -) 1-1-30 = 0 (1+5)(1-6)=0 so, The eigen values are = x= 3, -5,6 where x = 6 dq=3 13--5

eigen value. -8 Now, we have to find eigen vectors corresponding to each for x = 6 -4 -5 24 2 Reduced the above matrix in row-reduced echelon formi Ra R2- (1 - RI and R3 R3+(12) Ri 301-06 8 4 2 -4 3/2 0 Rat 9 R-1) RiRi - R2 -8 112 -30 TO -4 3/2 0 O 0 Sto, to x,=6isy eigen vector corresponding o -1/16 -38 GJA o o X3 is arebitary so het, X3=t JE - VIG 318 t where, az=t 2a x3 Now, for X2=3

E603 again by row- reduced echelon form of the R2 4 R2-(3) Ri above matrix. 5-5-4 2 -2 2 R3 R3 + (1/2) RI 4 2 2. - 5 6 -215 Nel sin? 0 -615 RE - R3-(3) R2 5 -4 2 6 -2/ 6/5 O o R1-100) 4/5 -215 2215 615 o 0 Rp Rp + 2 R2 R3-R2/) - 2 -215 2 615 " 0 -3 0 O o o 0 o 2 O =) eigen vectors - 3 O 10] o 0 0 0 23 is arbitowy rhet xy=+ t where x3=t -)

and for X₂ -5 =) [ 3 4 2 23 6 2 2. 4 1:10] x3 again by reduced echelon form 2013 - 3 2 -4 6 2 4 2. R3-6) R1 R2 y Rat 3 ) 2 -4 3 22 3 3 22 . 3 R3 R3-(3) R2 O 10 10 을 0 0 Now, R2 R2 ( To 3 2 1 - 0 0 o 6. R, TR1+ 4 R2 => MO — 0 0 0 O RI RI ( 135 =) 2 O O O 0 eigen vectors - [:]$0 X3 is webitowy &z=t t -) where, azot 3]