Question

QUESTION 2 The Gaussian elimination changes At = b to a row reduced form Rc =d. Now it is known that the complete solution of the system is --(3-(1) - (a) What is the 3 by 3 reduced row echelon matrix R and what is d? (b) Determine the rank and nullity A. (c) If the process of elimination subtracted 3 times row 1 from row 2 and then 5 times row 1 from row 3, what matrix connects R and d to the original A and b? Use this matrix to find A and b.

Answer 1

Answor Given data- The Gaussian elimination changes Ax=b to arow : RX=d ocduced from Rus d. Tx=rld The complete solution of the system is @ 14 2 YE D tc tů PX 4+2 G +502 otc to 0+0G + C2 ㅋ - 4 25 10 10 From equations ① and 2 we get, 4 25 RE 0 & d = رات - Cz OO .: R=(R-1) adj (R-1) (: A = 1 / adu (A)) JAI 4-9 -5 RE -| D 0 0 4

2 R= -% -514 0 e do b To determine the rank and nullity Ä Rank(A) = number of Pivot elements in reduced row echelon foom of A". nombor of non - Zedo rows of R. .: Rank (A):3 NOW Foom RONK - nullity theodem Rank (A) + nullity (A) = number of coulmns in 'À". ㅋ b+ nulity (A)=3 nullity (A)=0 AX=b RX=d Whate "R" is dow deduced foom Of A dis obtained from '' through some elementary Operations that are applied on A.

The Augmented matix is M = [nib M ~[Rid] The Operations are applied R2 R₂-3R, R3 R3-5R1 We need to apply following Obedations ose R2 R2+3RI R3> R375R, Convert [Ria] into [a b] is TO lo 0 3 10 5 0 :. [Aib] O - 2 -5/4 3 5 0 0 O -1/2 - -514 - 15/4 3 -12 3+1; 5/2 -2014 5+62 =514 - As 3 %. 15/4 & b= 3+1 5 -57% -]y 5+62