Geometry and Discrete Mathematics (Matrix)

[Cyto]

Hey all, I'm having some problems with this one homework question... We just did The Intersection of Three Planes using The augmented matrix... and here's my question...

For what value of k will the following set of planes intersect in a line?

x - 2y - z = 0
x + 9y - 5z = 0
kx - y + z = 0

 

[HallsofIvy]

Try solving the thre simultaneous equations. The solutions will, of course, involve k. Find a value of k for which you do NOT get a single value (that would probably be a value of k that makes a denominator 0.)

 

[M98Ranger]

To find the value of k, we can set up an augmented matrix with the coefficients of the variables and the constants on the right side. The matrix would look like this:

[1 -2 -1 | 0]
[1 9 -5 | 0]
[k -1 1 | 0]

From here, we can use row operations to transform the matrix into reduced row echelon form. This will help us find the value of k that will make the system of equations consistent and have a solution (intersect in a line). After performing row operations, we get the following matrix:

[1 0 -1 | 0]
[0 1 -1 | 0]
[0 0 (k+2) | 0]

To have a consistent system, the last row must have all zeros except for the last column, which represents the constant. This means that (k+2) must equal 0, and therefore k = -2. Therefore, the set of planes will intersect in a line when k = -2.