1)The matrix A is defined by
(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
(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.
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- Thread Starter
- 21-10-2006 15:48
- 22-10-2006 17:08
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 |
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.
- Thread Starter
- 23-10-2006 15:24
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.
- 23-10-2006 15:30
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.