Determine the value(s) of h and k such that the given system has no solution, a unique solution, many solutions.

Determine the value(s) of h and k such that the given system 

`x + 5y - 4z = 3,`

`3x + 12y - 12z =  - 3,`

`2x + 7y + hz = k`

has 
1. no solution,
2. a unique solution,
3. many solutions.

Answer:

        Given system

`x + 5y - 4z = 3`

`3x + 12y - 12z =  - 3`

`2x + 7y + hz = k`

        Now, in the matrix form

        let   

        So,

`AX = B`

        Now, in the form of Augmented matric

        As

`C = [A:B]`

        Now, their consistent matrices are

        Now, discuss the given conditions:

        i. NO SOLUTION

            `If  h =  - 8 and k \ne  - 6`

            then `the\[R(A) < R(B)`, there will be no solution.

            For that R(A)=2 and R(B)=3.

        ii. UNIQUE SOLUTION

            `If  h \ne  - 8 and k \ne  - 6`

            then the R(A) = R(B) = 3 = no. of unknown, there will be unique solutions.

        For that R(A)=3 and R(B)=3.

        iii. MANY SOLUTION

            If `h =  - 8 and k =  - 6` 

            then the R(A) = R(B) = 2 < no. of unknown, there will be many solutions.

            Like that R(A)=2 and R(B)=2.