Question

- Consider the matrix equation At = b given by the system 11 2 11 21 + 2:12 + 4.12 + 2.62 13 - 314 = b + 204 = by 13 + 5x4 = 63 + a) Write down the corresponding augmented matrix ( Ab) and use row operations to transform it into a matrix of the form (A b') where the coefficient matrix A' is in reduced row echelon form. (That is, you don't need to put the whole augmented matrix into rref). b) What condition (if any), in terms of the values b1,b2,b3, determines whether or not the reduced row echelon form of ( Ab) will have a pivot in the final column? c) For which values of (b1,b2,63) does the system have a solution? d) Solve the system in the case that (b1,b2, b3) = (1, 2, 1). Express the solution set in terms of a particular solution and a basis for the solution space of the associated homogeneous system.

Answer 1

۳ 63) 2. A=12 م م 4 o Que. Solution (C). Consider the matrix equatim , An=b given by the system, H1 +2012 -H3-304 = bi 201+ 482 tony =b2 M. +202 + 03 +5 814 = 63 ielt 2 -.-37 rain 12 4. o.21 m2 L 2. 1 5J La ri 2 -1 -3 and b= l bol II 2 I 5) ... 163 € C 1 2 - -3;b,] then augmented matrix [A16] = 1 2 ä ó 2 i byl 2 1 5 i b3 Le is uit TL 2--3 i bi] áo 2 : bz 1 2 1 5 ; 63 si 2 -1 - 3 (RS=R3-R2) = m 2 4 0 2 ; b2 | 13 -^3"^2) Lo 0 2 8 ; bg-bu 2 -3 ibi Ta 1o 2 8 169-26, (R} = R2-2Ro) lo 0 2 8 ; bg-6, J n 2 -3 i bi 7 Too 2 8 brazbi (R3 = R3-R2 To 0 0 0 ; bg - byt by ie (Alb] = 1 2 d doo doo to

T1 2 -1 -3 ; b. 7 o o 14 ; bz_bi (R2 = £12) n La oo Alb'] = 10 0 1 4 : b2-6 | o : bg-batby) 6 lb le À lo o 0 1 and 1 419 ho bzbatby b). . Since reduced row echelon form of [A16] is, ri 2 -1 -3 ; b. A' 16' ] = lo i į 4.: 62-bi Lo 0 0 0; bg-b2toi] If bz-bz to, 5o then (Allb'] will not have pivot in the final column. . So, the condition is, bz-be to, to for (A l b' ] to have prust in the final column. FG. Given. An ab reduces to Alnab' 11 312 = 152-6, Too 1 411 MB roi -1 -377,7 ie ri 2 Lo [bg-batb, o o o[

Since Alni=b' have a solution of mumk (A') = rank ([a'16']) ie rank ( [A' 16'] ) = 2 ( rank CA') =2) which is possible if bg-bztb, 50 ie. '63-62+b=0' is required condition for the given system to have a solution. .(d) for barbi equation reduces to b3 7 Jb, 2 -1 -37 loi stood o = |b2-bi | bg-bathi 1:4 M2 0 0 ] Oy ära] LO 0 0 of My J which gives , MA+202-03 – 3 My FI-- and N3+4 hy to - 1. for ny = 8, equation ® gives 82 +48 =0 ie N₂=-40 Putting values of M₂ =-40, My FA in ③ we get M. +2912 +40 – 30 = 1 4, + 2 H2 to =2

5) 7+2012 31-8 take H₂=t then above equation reduces to Ni +2+ = 10 5) N = I-A-26 Therefore solution of the given system of equation ..Ax=b for b=lél is of the form, It rul [-27 reJottlo ray] nalo +81-ul where, o in i barticular folution and particula