Question

Find the values of a and b for which the following system of linear equations is (i) inconsistent; (ii) has a unique solution; (iii) has infinitely many solutions. For the case where the system has infinitely many solutions, write the general solution.

x + y + z = a

x + 2y ? z = 0

x + by + 3z = 2

Answer 1

The augmented matrix of the given linear system is A (say) =

1	1	1	a
1	2	-1	0
1	b	3	2

To solve the given linear system, we will reduce A to its RREF as under:

1.Add -1 time row 1 to row 2

2.Add -1 time row 1 to row 3

3.Add -1 time row 2 to row 1

4.Add (1-b) times row 2 to row 3

5. Multiply row 3 by 1/2b

6.Add -3 time row 3 to row 1

7.Add 2 times row 3 to row 2

Then the RREF of A is

1	0	0	(ab+6a-6)/2b
0	1	0	2(1-a)/b
0	0	1	(a/2)-(a/b)+(1/b)

(i). In view of the 5^th row operation and RREF of A, the given linear system will be inconsistent if b = 0. (ii).In view of the 5^th row operation and the RREF of A, the given linear system has a unique solution if b is not equal to 0.

(ii). In view of the RREF of A, the given linear system cannot have infinite solutions.