Matrices (Hard Bit)

Question 1

What can you use to solve simultaneous equations in 3 unknowns?

Answer 1

Matrices

Question 2

How can you use matrices to help solve simultaneous equations?

Answer 2

You need to create a matrix by simply using the constants in front of the unknowns in the simultaneous equations. Then you need to write: that matrix multiplied by a 3X1 matrix of x y z. Then use the following formula where v is a 3X1 matrix of the constants.

Question 3

What are the 2 possibilities of a system of linear equations in 3 unknowns?

Answer 3

It can be consistent or inconsistent

Question 4

What does it mean is a system of 3 linear equations are consistent?

Answer 4

This means that there are at least 1 set of values that satisfy all of the equations simultaneously (otherwise it is inconsistent)

Question 5

What does it mean is a system of 3 linear equations are inconsistent?

Answer 5

This means that there are no set of values that satisfy all of the equations simultaneously (otherwise it is consistent)

Question 6

When visualising the matrices of 3 simultaneous equations how many different configurations are there?

Answer 6

6

Question 7

When visualising the matrices of 3 simultaneous equations - what are the different configurations?

Answer 7

Planes meet at 1 point
A sheaf
Prism
Parallel planes
Parallel planes
Same plane

Question 8

What are the 2 phrases that you need to know?

Answer 8

Sheaf and prism

Question 9

When visualising the matrices of 3 simultaneous equations - what is a sheaf?

Answer 9

Question 10

When visualising the matrices of 3 simultaneous equations - what is a prism?

Answer 10

Question 11

When visualising the matrices of 3 simultaneous equations - walk through the imagery in the flow chart with few words

Answer 11

Question 12

When visualising the matrices of 3 simultaneous equations - explain the flow chart?

Answer 12

You need to check the detM
If detM doesn’t equal 0 (it is non singular) then it has to be: places meet at 1 point.

If DetM=0 then you need to check the consistency

If it is consistent then solutions do exist and it can either be a sheaf or same plane
If it is the same plane then all the equations will be scalar factors if not it is the sheaf

If it is non consistent then it is inconsistent
You then need to check how many of the planes are parallel. If 2 or 3 are parallel then it is parallel planes
If none are parallel then it is a prism

Question 13

How can you check the consistency of the equations?

Answer 13

Solving them either by using algebra or by using matrices

Question 14

How can you check for parallel planes?

Answer 14

By looking at the equations.

For equations with x+y+z=b

Parallel planes with a scalar multiple of x,y,z but not b, will be parallel with gaps between them

If x,y,z and b share a scalar multiple then they will overlap in the same plane

Question 15

Just learn how to do this question:

Answer 15

Question 16

Explain this structure?

Answer 16

The planes meet at a point. The system of equations is consistent and has one solution represented by this point.

This is the only case where the corresponding matrix is non-singular

Question 17

Explain this structure?

Answer 17

The planes form a sheaf.
The system of equations is consistent and has infinitely many solutions represented by the line of intersection of the 3 planes

Question 18

Explain this structure?

Answer 18

Question 19

Explain this structure?

Answer 19

Question 20

Explain this structure?

Answer 20