Question

4. Give examples of the following cases. Provide a reasoning for your answer. (a) (5 points) A basis B = {P1, P2, P3} of P, such that [t?]B (b) (5 points) Two 2x2 matrices A and B with the same determinant, but not similar. (c) (5 points) A square matrix A with all entries nonzero that has a two dimensional cigenspace.

Answer 1

tabe cal fc {1+2, t, -+ ] it b644 of ? ť? 1(1+43 ) + ot t1(1+z2 [f²] = [ cofficient of basis Linear combination of same] [:] こ Tai [" 139 Ć A b) A = و تا BE [: ;] - с TAI - 1 1B) = 1 eigen value of A eigen value of B are -1, -1 1 they have not 2 Same eigen value =) not similar SC?

(c ī Lot Az 2 -1 2 2 2 - 2 Find eigen value IA-h11-0 ーーん 2 2 2 2 2-2 -1 -1 2-h =) (-1-h) [2-4)2-1] -2 [2C2-W) + 2] +2 (-2-2C2-A))=0 El-h) [4th?_unal] 2 Lu-2h+2] + 2 [-2-4+2 h]=c =) El-h) La2_uht] -2 [6-2h] + 2 -2 [6-21] +2 (-6+24) = 0 z (-i-h) [12_4h+3] -127 46-12+unzo =) -42+46-3–437 442-3h -824 + 8h = 0 =) -23+3 +3h2 tah -27 = 3 h3_3h2 Gh+2720 3 -9 22 0-27 cal = (h-3)Ch2-9) 2 h = 3, 3, -3 Find eigen vector for ha 3 X is eigen vector (A-3I) X=0 .4 2 2 el 202 }= 2 -1 23 2 -1 -1 R27 R 24 RI R37 Ryt + ŹŹRI -4 2 2 71 1032-63 -431, +222 + 273 =) -2x =-12-13 - 2 +24,2 x2 + x3 pus 23 Carb) ciden Spare 7 LE})= a(?-) * **{*}} al que two dimensional eisen Sbare forhasan