Thread: Homogeneous system solution

I am in a matrices and power function class,

we are asked to determine if this homogeneous system has a unique solution without explicitly solving it.

-x + 2y - z = 0

2x + y + z = 0

x - y + 2z = 0

So, I setup the matrix and checked that the determinant does not equal 0. It does not, it is - 4. Then the question wants me to give the values for x,y and z without solving for them based on what I already know.

The values, it says, are x = y = z = 0

Can someone please tell me how I am to know this????
Thanks Frostking

Originally Posted by Frostking

I am in a matrices and power function class,

we are asked to determine if this homogeneous system has a unique solution without explicitly solving it.

-x + 2y - z = 0

2x + y + z = 0

x - y + 2z = 0

So, I setup the matrix and checked that the determinant does not equal 0. It does not, it is - 4. Then the question wants me to give the values for x,y and z without solving for them based on what I already know.

The values, it says, are x = y = z = 0

Can someone please tell me how I am to know this????
Thanks Frostking

Since the coeffient matrix is invertable you know the system has Exactly 1 unique solution. Homogenous systems always have the trivial solution. x=y=z=0.

Combining these two facts gives you what you want.

Putting these two