Matrices Year 2

Question 1

How would you find the intersection of 3 planes using matrices?

Answer 1

1) convert simultaneous equations into a matrix multiplication

(3x3 matrix of coefficients called matrix M) x (3x1 matrix of x and y) = (3x1 matrix of answers)

2) Premultiply both sides by inverse of M to find x and y matrix

Question 2

What are the 4 different arrangements of the three planes?

Answer 2

1) meet at one point
2) are all parallel
3) form triangular prism
4) 2 are parallel with one crossing the other two
5) a sheaf so all meet in one line

Question 3

If the determinant of the 3x3 matrix is not 0 what does that mean geometrically?

Answer 3

They all meet at one point and you can find this intersection using the method of inverse matrices.

Question 4

If the determinant of the 3x3 matrix is 0 what does that mean geometrically?

Answer 4

It is either:
sheaf  
triangular prism
3 parallel planes 
or 2 parallel planes and one crossing

You can not find the intersection as there is not one

Question 5

How would you distinguish between the other arrangements if |M| = 0?

Answer 5

If the 3 simultaneous equations are consistent then is it a sheaf.
If the simultaneous equations are inconsistent then it is either a triangular prism, 3 parallel or 2 parallel and one crossing.