Question

2. (a) The linear system 212 + bıy+cı2 = 0 02.2 + b2y + 0,2 = 0 232 + b3y + c32 = 0 has infinitely many solutions, i. (3 marks) What can you say about the number of solutions of the following system? Explain. 412+by+C12 = 3 22x + b2y + C22 azx + b3y + C32 11 = 7 bi ii. (3 marks) Is the set W b2 linearly independent? b3 Explain. (b) (4 marks) The augmented matrix for a linear system has been reduced by row operations to the given row echelon form. Solve the system. 10 1 2 1 2 0 1 1 3 -41 0 0 0 1 12

Answer 1

2) a) (i) Since the linear system 0 a11 +biy + cız 22x + b2y + c22 az1 + b3y + c3z 0 0 has infinitely many solutions. a b c Hence, For the Matrix A=anbz Cz a bz cz The rank is less than 3 Now for the system 3 2,2 +biy + 2 22x + b2y + coz azx + b3y + C32 7 11 Now two cases may arise q by a 3 Case-1) If Rank(A) = Rank(Ab) Where Ab=ay by Cy 5 a bz cz 7 Then there will be infinitely many solution Case-2) If Rank(A) Rank(Ab) Then there will be no solution (ii) Since Rank(A) <3 a b c Hence Det(A) = aby a b c =0 ai bi C1 So, the set W = b2 C2 is not linearly Independent b3 C3 b) The system 1 0 1 2 -12 has five variables and three equations. 0 1 1 3 -41 0 0 0 1 112 1 0 1 2 - 1 Here, The rank of the coefficient matrix 0 1 1 3 -4 is 3<5 0 0 0 1 1 and 1 0 1 2 -1 Rank of 0 1 1 3 -4 = Rank of 1 0 1 2 -12 0 1 1 3 -41 10 0 0 1 12 0 0 0 1 1 Hence there will be infinite solution. (1) Now the system can be written as X+z+2t-w=2 y+2+3t-4w=1 t+w=2 From (ii) w=2-t .....(iv) From (ü) and (iv) y+2+7t=9......(v) From(i) and (iv) x+2+3t=4....(vi) Hence(v)-(vi) gives y-x+4t=5..... .....(vii) From (iv), (vi) and (vii) The solution is (x,y,z,t)=(x,x +5-4t,-X+4-3,2-t)) = x(11-10)+1(0,-4,-3,-1)+(0,5,4,2) Where x and t are arbitrary real numbers.