Question

Investigate for what values of 𝝁 "𝒂𝒏𝒅" 𝝀 the equations : `2x+3y+5z=9`

`7x+3y-2z=8`

`2x+3y+λz=μ`

Have (i) no solution (ii) unique solution (iii) Infinite value 

Answer 1

Given eqn : 

`2x+3y+5z=9` 

`7x+3y-2z=8`

`2x+3y+λz=μ` 

    `  A x=B`

∴ `[[2,3,5],[7,3,-2],[2,3,λ]] [[x],[y],[z]] =[[9],[8],[μ]]` 

Argumented matrix is : `[[2,3,5],[7,3,-2],[2,3,λ]][[9],[8],[μ]]`

`R_3-R_1`  

→` [[2,3,5],[7,3,-2],[0,0,λ-5]]  [[6],[4],[μ-9]]`

(i) When 𝜆=5,𝝁≠𝟗 𝒕𝒉𝒆𝒏 𝒓(𝒂)= )=𝟐,𝒓(𝑨⋮𝑩)=𝟑
r(A)≠𝒓(𝑨⋮𝑩)
No Solution.
(ii) When 𝜆≠5,𝝁≠𝟗 ,𝒓(𝑨)=𝒓(𝑨⋮𝑩)=𝟑
Unique solution exist.
(iii) When 𝜆=5,𝝁=𝟗 𝒓(𝑨)=𝒓(𝑨⋮𝑩)=𝟐<𝟑
Infinite solution.