Question

(33 pts) This question is about the matrix = ſi 2 [3 2 0 4 1 6 3 1] 4 9 co (a) Find a lower triangular L and an upper triangular U so that A = LU. (b) Find the reduced row echelon form R = rref(A). How many independent columns in A? (c) If the vector b is the sum of the four columns of A, write down the complete solution to Ax = b

Answer 1

- Az / 120 l 1 2 4 1 a 1 3 6 3 9.) ca? First we find the Row rede simple echelon form of - A to get u. A /120 l R 1 / 2 oll 2414 - loo 12 1 3 6 39 R3-3R, Loo 36) | R3-322 1 1 2 0 11 Oo 2 21 ol No let La llu o LI20 I Claro oli 2011 24 14 lal 121 lze bez o o o 121 Oo oo 3 6 3 9 , 1 bei dir les ! all, 211, o lli 1 (1, 21, , 14, +21, - Herre lied, do 1 = 2, 13,2 3., dzz =d, l32 = 3 CSscanenoddvith denscanner Csay)

Hence La / loo 1 2 1 0 - -0 oo i "Oo oo (b). We already have the simple echelon form I - Pirots. O O O 2 Clearly we can see this is the Row-Reduced echelon form. As there are 2 pivots. there are. only 1st and 3rd column.ie 2 column's are. Linearly independent. - 2 1 ro «2 let bo ($)+($)+()*(3) 21 1. We apply the row operations on the. Augmented matrix. /2 olla I R2-28, 2o14. 1 2 4 14 →lo 0 1 2 3. - 1 3 6 39 21 ). Rz 3R, loo361 | R₂-322. I 2 0 1 oo 1 2 0 0 0 0 Eo Imo canned with CamScanner

We have, rank (A) a rank (Alb). Hence solution of Anab exists. If xa I is a solution of Anab then We have by the Row reduction. *,+222 +24=4. Here CS$canned with Camstagnet 274 2 3. agglq are deponclent - variable

2,2 3-224. = 4-24-14. Hence / 1 / 9-222-da ) 3-209 ** ° °("*") (9)-() *- (1)-(1)-(3) CSER Heure The complete solution of Anabis. owo CScanned with CamScanner