Question

Find a 3-vector xo for which Axo Xo and Xo has length 1. b. Compute A"xo for n= 1,2.. 7. A square matrix A is said to be idempotent if AA- A. an canc cellation The a. Show that the following matrix is idempotent: 2.40 2-2 -4 (G3 -1 241) b. Show that if AB= A and BA- B, then A and B are both idempotent. Show that if A is idempotent, then A" = A for all positive integers n. -2 A2) eas Harder Problems c d) 8. Suppose A = . Show that A2 = (a + d)A - (ad - bc)I2. b. Use part (a) to show that A3 the equality in part (a) by A, and use the equality A = 0 to derive an equation, which you should then multiply by A once again.) c. Give an example of a matrix A such that A A0, but A #0. 0 implies A2=0. (Hint: Multiply

solve 7th and 8th

Answer 1

12 -2 -41/2 -4 -41 1 3 4 11- 3 41 -2 -3/1 i -2 -3 7 442-4 1-2-3+4 1 2+2-3 -4-6+8 2 +9-8 -2- 6+6 -8-8+12) 4 +12-12 -4 -8+9] = A 1-1 II 3 -2 4 -3) A² = A. 1. Hence A is idempotent. (b.) het AB = A and BA=B. A.A= (AB). (AB) = A (BA)B by associabive law' = (AB) B. (BA=B) - AB (AB=A) = A .: AAA. Hence A is idempotent fimilovely BiB = (BA). (BA) - BIAB), A - (BA). A I B.A B.B=B Herve B is idempotent

(6.) fuppose that A is idempotent then A²-A: We have to show that AM-A for all nyt For, n=1,2 A= A. fet k be a positive integer fuch that Ak = A Them Akt! – ALA = A:A-ASA - Akt=A Heure by induction we have A - A for all ny, l. . 4-ASA (B) la) A-la " 4² = ( a bla bl=/ a²t be abtbd.) I called Tearde ebrd?) 1 a²rad as tb di (atd) A = 1 I cared ad+d² (ad-be) I, 1 ad-be ol o ad-bc) la+d) A -(ad-be) kn = aftbe abt.) A2 I catcol d²tbel

(b.) fuppose that A²=0 :: A² - urd) A - ( ad-bye) 12 - 13 - kra) Ae - led-se) =) Card) a-lad-be) A = 0 0 =)latel) A³ - led-sy) as - 0 | => - (adsca=0 s a= 0 A=18] Then A²=A²=0 But A to