Solving systems of equations using matrices

I’m doing question 6

And I’m really stuck on identifying the geometric configuration of the planes part.

These are my notes on singularity and consistency

But they don’t help.

There are answers in the book

There’s no explanation as to how it’s known there are an infinite amount of solutions or no solutions.

What am I missing please.

Thanks.

Reply 1

mqb2766

Both i) and iii) correspond to det=0 which means that the third row in the matrix is a linear combination of the other two (for instance). Also the matrix only depends on the left hand side of the 3 equations so the x,y,z coefficients. So in 2d think of train lines which are either parallel and head off to infinty never meeting or are on top of each other and they represent the same line. so you nned to look at the right hand side (the y intercept in 2d) to determine which you have.

iIn i) q=-10 you should see that
2* frist equation - 6* second equation = third equation
So the right hand sides are consistent as -32+30=-2 So really you have two equations (the third doesnt add anything) so the solution is a line (First two planes intersecting in a line and that line also lying in the thrid plane). Its singular and consistent as there there is a line of solutions which satisfy all three equations.

in iii) the left hand sides (x,y,z part) satisfy
- first equation + 3*second equation = 5*third equation
However, this does not saisfy the right hand sides as 16+3/2 = 17 1/2 not -10 So the equations are inconsistent and you have the third plane being offset from the line of intersection of the first two and never meeting (paralllel train tracks). Its singular and inconsistent as there are no solutions which satisfy all 3 equations.

(edited 9 months ago)

Reply 2

maggiehodgson

Original post by mqb2766

Both i) and iii) correspond to det=0 which means that the third row in the matrix is a linear combination of the other two (for instance). Also the matrix only depends on the left hand side of the 3 equations so the x,y,z coefficients. So in 2d think of train lines which are eitehr parallel and head off to infinty never meeting or are on top of each other and they represent the same line. so you nned to look at the right hand side (the y intercept in 2d) to determine which you have.

iIn i) q=-10 you should see that
2* frist equation - 6* second equation = third equation
So the right hand sides are consistent as -32+30=-2 So really you have two equations (the third doesnt add anything) so the solution is a line (First two planes intersecting in a line and that line also lying in the thrid plane). Its singular and consistent as there there is a line of solutions which satisfy all three equations.

in iii) the left hand sides (x,y,z part) satisfy
- first equation + 3*second equation = 5*third equation
However, this does not saisfy the right hand sides as 16-3/2 = 14 1/2 not -10. So the equations are inconsistent and you have the third plane being offset from the line of intersection of the first two and neve meating (paralllel train tracks). Its singular and inconsistent as there are no solutions which satisfy all 3 equations.