FP3 matrices

Hi

I'm told that three planes have the following equations:



Find the possible values of the constant k for which the equations do not have a unique solution.

I don't really understand what they're asking for. Here's the worked solution for it (Q 8) but I'm afraid this doesn't help me much.

I understand that they've made the three equations in a singular matrix but why does this tell you the values for when "the equations do not have a unique solution"?

In summary, I don't understand the implications on the three equations if the value of k is $1$ or $-\frac{2}{3}$ (i.e. the values where the matrix of the three equations is singular).

I don't think this is strictly on my syllabus (I can't find it in any of my textbooks) but it'd be nice to understand it anyway.

Thank you

Hi

I'm told that three planes have the following equations:



Find the possible values of the constant k for which the equations do not have a unique solution.

I don't really understand what they're asking for. Here's the worked solution for it (Q 8) but I'm afraid this doesn't help me much.

I understand that they've made the three equations in a singular matrix but why does this tell you the values for when "the equations do not have a unique solution"?

In summary, I don't understand the implications on the three equations if the value of k is $1$ or $-\frac{2}{3}$ (i.e. the values where the matrix of the three equations is singular).

I don't think this is strictly on my syllabus (I can't find it in any of my textbooks) but it'd be nice to understand it anyway.

Thank you

If you were just solving linear equations (forgetting about the geometry) then you could rewrite them as a matrix equation. If the matrix of coefficients had an inverse, you could multiply both sides by that inverse to get a unique solution for x,y,z.

However, if the equations represent planes, we can't have a unique solution - because 3 planes can't meet at a point (try it with 3 sheets of cardboard if you don't believe me!). Therefore the matrix can't have an inverse i.e. it is singular.

The rest of it is just sorting out whether all 3 planes meet in a single line, or meet in pairs at different lines, or 2 (or more) are parallel to each other.

When the matrix is singular, there is no unique solution.

Do you know how to find the solution of a set of simultaneous equations by using matrices?

If you were just solving linear equations (forgetting about the geometry) then you could rewrite them as a matrix equation. If the matrix of coefficients had an inverse, you could multiply both sides by that inverse to get a unique solution for x,y,z.

However, if the equations represent planes, we can't have a unique solution - because 3 planes can't meet at a point (try it with 3 sheets of cardboard if you don't believe me!). Therefore the matrix can't have an inverse i.e. it is singular.

The rest of it is just sorting out whether all 3 planes meet in a single line, or meet in pairs at different lines, or 2 (or more) are parallel to each other.

Actually, sorry, but I can't quite see why three planes can't meet at a point. For example, if you look at the first diagram in this link, it looks like they meet at just a point. I know I'm missing something here but can't quite see what.

Thank you

If you were just solving linear equations (forgetting about the geometry) then you could rewrite them as a matrix equation. If the matrix of coefficients had an inverse, you could multiply both sides by that inverse to get a unique solution for x,y,z.

However, if the equations represent planes, we can't have a unique solution - because 3 planes can't meet at a point (try it with 3 sheets of cardboard if you don't believe me!). Therefore the matrix can't have an inverse i.e. it is singular.

Actually, sorry, but I can't quite see why three planes can't meet at a point. For example, if you look at the first diagram in this link, it looks like they meet at just a point. I know I'm missing something here but can't quite see what.

Thank you

Three planes can indeed meet at a point - consider the perpendicular planes x=0, y=0, z=0, which meet at the origin as the most obvious example. And indeed this is the condition for 3 linear equations having a unique solution - if the planes they represent meet at a point, there is only one point in 3-space that satisfies them all, so it is unique.

Yeah, sorry, I've always had trouble with 3 dimensions

Three planes can indeed meet at a point - consider the perpendicular planes x=0, y=0, z=0, which meet at the origin as the most obvious example. And indeed this is the condition for 3 linear equations having a unique solution - if the planes they represent meet at a point, there is only one point in 3-space that satisfies them all, so it is unique.

3 planes can meet in a point. 2 planes however, cannot.