1. (a) (8 marks) Find the general solution of the following system of linear equ
ID: 3138637 • Letter: 1
Question
1. (a) (8 marks) Find the general solution of the following system of linear equations using elementary row operations on the augmented matrix 2r 572 6x3 + 2x4 = 3r 4r2 8x3 4 (b) (6 marks) Determine whether the following system of linear equations is consis- tent: x2 + 2x3 = 1 2x1 + 2x2 + x3 = 3 3r1 + x2 + 3x3 = 5 (c) (6 marks) For each of the following statements, declare whether it is true or false (i) A system of 3 linear equations with 4 variables always has infinitely many (ii) Every matrix can be converted to a row echelon matrix by a sequence of (iii) Elementary row operations on an augmented matrix may change the solution solutions elementary row operations set of the associated system of linear equationsExplanation / Answer
1(a). The augmented matrix of the given linear system is A =
2
5
-6
2
3
1
2
-3
-1
2
3
4
-8
-2
-1
To solve the given linear system, we will reduce A to its RREF as under:
Multiply the 1st row by ½
Add -1 times the 1st row to the 2nd row
Add -3 times the 1st row to the 3rd row
Multiply the 2nd row by -2
Add 7/2 times the 2nd row to the 3rd row
Add 3 times the 3rd row to the 1st row
Add -5/2 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
18
-23
0
1
0
4
-1
0
0
1
9
-9
Thus, the given linear system is equivalent to x1+18x4 = -23 or, x1= -23-18x4, x2+4x4= -1 or, x2 = -1-4x4 and x3+9x4=-9 or,x3=-9-9x4.Now,let x4= t. Then (x1,x2,x3,x4) =(-23-18t,-1-4t,-9-9t,t) = (-23,-1,-9,0) +t(-18,-4,-9,1) where t is an arbitrary real number.
(b). The augmented matrix of the given linear system is A =
1
-1
2
1
2
2
1
3
3
1
3
5
The RREF of A is
1
0
5/4
0
0
1
-3/4
0
0
0
0
1
Since 0 cannot be equal to 1, hence the given linear system is inconsistent.
(c ).(i). False. The 4 linear equations in 3 variables should, additionally, be linearly independent.
(ii). True. Every matrix can be converted into row echelon form ( not matrix) by elementary row operations.
(iii). False.
2
5
-6
2
3
1
2
-3
-1
2
3
4
-8
-2
-1
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