Question

Consider the homogeneous linear system 1 +3y + 4z=0,21 +22=0,-y-z=0] Give the coefficient matrix for this system: b sin (a a ar 00 22 Give the augmented matrix for this system: ab sin(a) 00 a Reduce the augmented matrix to reduced row-echelon form: a ab sin (a) f 8 a 12 ОТ Give a basis for the set of all solutions of the system. Syntax: Enter your answer as a set of vectors in one of the following forms (depending on the number of vectors in your basis): {} {< 41, 42, ug >} {< 41, 42, ug >,< 01, 02, 03 >} {< 41, 42, ug >,<01,U2, U3 >,< 21, 22, wg >}

Answer 1

Page Considee the homogeneous linear system x+3y +43=0 2x + 2z = 0 The above system of equations is of the form АХЕО where coefficient mateix is, A = 1 3. old 2 - - & Augmented mateix is [Alo]: 6 2 2 O Io o -) O R3 <1 R2 [Alo] 3 O - 2 2 R3 R3 -2RI [Alo 1 3 G - 1 - 1 0 d - 6 - 6 R3 R3-6R2 [Alo 1 3

Thus Eank of [Alo] =2 & no. of unknowns = 3 (no of non-zero rows -Cie x, y, 3) Thus eank of Calo] = 2 +3= no of unknowns. sank of Calo] < no. of yoknowns le system has infinite solution Now AX=0 becomes 3 ما - -) У 3 0 o D $ x+3y + 43=0 -y-3=0 y+3=0 y = -31 Put y=-3 in x+3y + 43=0 x-33 +6850 IT & TM x + 3 =0 = Tx=-3 Here no of independant vociable cank of unknowns 3 - 2 = 1 (i-e 3) Put る二十 y=-3 = -t f x=-3= -t Thus solin of the system is y = -t 3=t ie s-t - t t x = -t - 1 Thus soln is < (-1,-1,1)>.
the solution of given system of equations is <(-1,-1,1)>