Question

Investigate for what values of μ and λ the equations x+y+z=6, x+2y+3z=10, x+2y+λz=μ has
1) No solution
2) A unique solution
3) Infinite number of solutions. 

Answer 1

we have `[(1,1,1),(1,2,3),(1,2,lambda)] [(x),(y),(z)]=[(6),(10),(mu)]`

by `R_2-R_1,    R_3-R_2`

`[(1,1,1),(0,1,2),(0,0,lambda-3)] [(x),(y),(z)]=[(6),(4),(mu-10)]`

i) The system has unique solution if the coefficient matrix is non-singular (or the rank A, r = the number of unknowns, n=3).
This requires λ-3 not equal to 0, Hence λ is not equal to 3.
Hence the system has unique solution.

ii) If λ=3 the coefficient matrix and the augmented matrix becomes

`[(1,1,1),(0,1,2),(0,0,0)]` and `[(1,1,1,6),(0,1,2,4),(0,0,0,mu-10)]`

The rank of A=2 the rank of [A,B] will be also 2 if μ=10.
Thus if λ=3 and μ=10 the system is consistent. But the rank of A (=2) is less than the number of unknowns (=3). Hence the equation will posses infinite solutions.

iii) If λ=3 and μ≠10, the rank of A=2, and the rank of [A,B]=3. They are not equal and the equations will be inconsistent and will not posses any solution.