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    1)The matrix A is defined by
    A=
    (1 3 2)
    (1 -1 -1)
    (2 2 P)
    Find the rank of A, distinguishing between the cases "P=1" and "P is not equal to 1".

    My echelon form is
    A=
    (1 0 -1/4)
    (0 1 3/4)
    (0 0 P-1)


    This is okay, and so I know the rank for both cases, but I have problems in the next few parts.

    Consider the system S of equations:
    x + 3y + 2z = 1,
    x - y - z = 0,
    2x + 2y + Pz = 3P + K -2,

    i)Show that if P is not equal to 1 then S has a unique solution. Find this solution in the case K=0.
    ii)Show that if P=1 and K=0 then S has an infinite number of solutions.
    iii)Show that if P=1 and K is not equal to 0 then S has no solution.

    Can someone tell me the steps to do these three parts, because I'm new to this topic and I really don't know where to start for this question.

    Thank you.
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    If P≠1, then all the 3 rows are independent as they are not linear combination of each other therefore it has a unique solution as these planes will cut at a single pt.

    U can sove the 1st part by applying the row operation to the augmented matrix an example is given below

    | 1 3 2 | 1 |
    | 1 -1 -1 | 0 |
    | 2 2 P | 3P-2|

    my matrix reduced to

    | 1 0 -1 | 1 |
    | 0 1 0 | 1 |
    | 0 0 P+2 | 3P-6 |

    therefore z(P+2)=3P-6

    z= (3P-6)/(P+2)
    y=1 and;
    x= (4P-4)/(P+2)

    for the 2nd, apply the same procedure and u will not get a unique solution, as these planes will intersect on a common line; just try it!

    Plot these equations in graphing software in order make the question clear.

    i'll be solving 3rd part sometime later, if wud have time that is.

    I hopw that the sol. is clear to u.
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    Thank you.

    Actually what is meant by "unique solution"?

    I kind of know how to do the questions, but can you tell me what is the "condition" that "S has no solution" and "S has an infinite number of solutions"? I meant from what we can judge it.

    Thank you.
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    If you have 2 equations in 3 unknowns, i.e.

    x + y + z = 21
    3x - y +7z = 4

    Then you have 3 unknowns in 2 equations then you infinitely many solutions since you let one unknown be a constant, say b then write x, y, z in terms of b.

    You have unique solutions if you have say x = 2, y = 1 and these are the only solutions.

    You have no solutions if you get 1=0 when solving the system of simultaneous equations.
 
 
 

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