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# Matrices watch

1. 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.
2. 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.
3. 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.
4. 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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Updated: October 23, 2006
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