Find all values of k for which the linear system x 3y 2z

Find all values of k for which the linear system {x + 3y + 2z = 4 2x + 7y + 7z = 0 3x + 10y + k^2 z= k a) Has a unique solution; b) Has infinitely many solutions; c) Has no solutions.

Solution

The given linear system can be represented in matrix form as AX = b, where A is the coefficient matrix and b = ( 4,0,k)^T. The augmented matrix of the given linear system is B = [A,b]=

1

3

2

4

2

7

7

0

3

10

k²

k

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

1.Add -2 times the 1st row to the 2nd row           

2.Add -3 times the 1st row to the 3rd row

3.Add -1 times the 2nd row to the 3rd row

4.Multiply the 3rd row by 1/(k²-9)

5.Add -3 times the 3rd row to the 2nd row

6.Add -2 times the 3rd row to the 1st row

7.Add -3 times the 2nd row to the 1st row

Then the RREF of B is

1

0

0

(28k²+7k-280)/(k²-9)

0

1

0

(-8k²-3k+84)/ (k²-9)

0

0

1

(k-4)/ (k²-9)

(a). It may now be observed that x = (28k²+7k-280)/(k²-9), y = (-8k²-3k+84)/ (k²-9) and z= (k-4)/ (k²-9). Since division by 0 is not defined, hence if k²-9 0, i.e. if k ± 3, then the given linear system has a unique solution.

(b). In no case, will the given system have infinitely many solution.

(c). If k = ± 3, then the given linear system is inconsistent as in both these cases, the RREF of B is

1

0

-7

0

0

1

3

0

0

0

0

1

Since 0 1, the given linear system will become inconsistent.

1	3	2	4
2	7	7	0
3	10	k2	k