Core Pure - Matrices (Hard Bit) Flashcards

1
Q

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

A

Matrices

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2
Q

How can you use matrices to help solve simultaneous equations?

A

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.

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3
Q

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

A

It can be consistent or inconsistent

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4
Q

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

A

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

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5
Q

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

A

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

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6
Q

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

A

6

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7
Q

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

A

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

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8
Q

What are the 2 phrases that you need to know?

A

Sheaf and prism

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9
Q

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

A
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10
Q

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

A
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11
Q

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

A
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12
Q

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

A

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

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13
Q

How can you check the consistency of the equations?

A

Solving them either by using algebra or by using matrices

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14
Q

How can you check for parallel planes?

A

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

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15
Q

Just learn how to do this question:

A
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16
Q

Explain this structure?

A

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

17
Q

Explain this structure?

A

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

18
Q

Explain this structure?

A
19
Q

Explain this structure?

A
20
Q

Explain this structure?

A