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zA Level Further Maths > Core Pure - Matricies > Flashcards

Core Pure - Matricies Flashcards

1
Q

What is a matrix?

A

An array of elements set out in a pair of brackets

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2
Q

How cn you describe the size of a matrix?

A

By the number of rows and columns
Generally you say the number of rows then columns

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3
Q

What is a row?

A

The number of lines of horizontal elements

1
2
3

(Rows go down)

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4
Q

What is a column?

A

The number of vertical lines of elements

1 2 3

(Columns go right)

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5
Q

Give an example of a 2X4 matrix?

A
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6
Q

What is a square matrix?

A

A matrix with the same number of rows and columns

Eg 1x1 or 2x2 or 3x3

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7
Q

What is a zero matrix?

A

A zero matrix is a matrix in which all of the elements are zero

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8
Q

How is a zero matrix denoted?

A

0

)dbdhdbcwidn odjn ik

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9
Q

What is an identity matrix?

Give an example?

A

A square matrix where all the elements on the leading diagonal are 1 and the rest are 0

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10
Q

What is something that you dan say about matrices that are the same size?

A

They are said to be additively confortable

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11
Q

How do you add or subtract matrices?

A

You just + or - each corresponding element

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12
Q

What needs to be the case for you to be able to add 2 matrices?

A

They need to be the same size
(They need to be additively confortable)

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13
Q

How can you multiply a matrix by a scale factor?

A

You just multiply each element by the scale factor

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14
Q

How can you divide a matrix by a scale factor?

A

You need to multiply by the inverse of that

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15
Q

What is a scale factor?

A

A number

Eg 1,5, sqr(2), e

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16
Q

What needs to be the case for matrix multiplication?

A

The n.columns in the first matrix is equal to the n.rows in the second matrix

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17
Q

What size will a matrix be after matrix multiplication?

A

The same n.rows as the first matrix and the same n.columns of the second matrix

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18
Q

MatA has size: nxm
MatB has size: mxk
What size will MatC have?

A

nxk

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19
Q

If you CAN multiply MatA with MatB, what can be said about these two matrices?

A

If AB exists then MatA is said to be multiplicatively confortable with MatB

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20
Q

Multiply these matrices:

A
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21
Q

What is very important to remember with matrix multiplication?

A

In general AB doesn’t equal BA

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22
Q

What are the main take aways from matrix multiplication?

A

In general AB doesn’t equal BA
If AB exists BA doesn’t necessarily exist

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23
Q

What heeds to be true for a matrix to have a determinant?

A

It must be a square matrix

24
Q

What is the determinant?

A

It is a scalar value associated with that matrix

25
Q

How do you calculate the determinant of M? What is an important fact about this matrix that allows us to calculate the determinant like this?

A

DetM=ad-bc

It is important that it is a square matrix as non square matrices don’t have determinants (at least in terms of this spec)

It is important that it is a 2x2 matrix as you need a different method to calculate the determinant of a 3x3 matrix

26
Q

How is the determinant or a matrix, M, often written as?

A

DetM or

27
Q

If DetM=0 what do you know about the matrix?

A

It is a singular matrix and it doesn’t have an inverse

28
Q

If DetM doesn’t equal 0 then what do you know?

A

It is a non singular matrix and it will have an inverse

29
Q

What is a singular matrix?

A

A matrix with a determinant equal to 0

30
Q

What is a non singular matrix?

A

A matrix with a determinant of not equal to 0

31
Q

What do you need to know to know to find the determinant of a 3x3 matrix?

What is the general aim of this step?

A

The formula:

The general aim is to reduce the 3x3 matrix to a 2x2 martrix

32
Q

Explain a bit about the formula to find the determinant of a 3x3 matrix?

A

It is designed to reduce the 3x3 matrix into a 2x2 form
Each of the elements along the top row are multiplied by its minor

33
Q

What is the minor of an element in a 3x3 matrix?

A

It is the determinant of the 2x2 matrix that remains after the row and column containing that element have been crossed out

34
Q

What are the main things that you need to be able to do with matrices?

A

Multiply a matrix by another matrix
Find the determinant
Find the inverse
Transpose a matrix

May be more in final topickjdncjidncojsdnjsk (

35
Q

What is an important thing to remember when inverting a 2x2 matrix?

A

You can only inverse a 2x2 matrix which is non singular (DetM doesn’t equal 0)

36
Q

How do you calculate the inverse of a matrix?

A
37
Q

Why can’t you find the inverse of a singular matrix?

A

A singular matrix had a determinant of 0 so 1/DetM = 1/0 which is undeffined

38
Q

What is the general notation of the inverse of MatM?

A
39
Q

What is the general notation for the transpose of MatM?

A
40
Q

What is a transpose of a matrix?

A

It is found by interchanging the rows and columns

41
Q

How many steps are there to finding out the inverse of a 3x3 matrix?

A

5

42
Q

What is the first step of finding the inverse of a 3x3 matrix?
(Find the inverse of MatA)

A

Find DetA

43
Q

What is the second step of finding the inverse of a 3x3 matrix?
(Find the inverse of MatA)

A

Find the matrix of minors of MatA
This involves replacing each of the 9 elements of MatA by its minor
You will end up with a normal 3x3 matrix

44
Q

What is the third step of finding the inverse of a 3x3 matrix?
(Find the inverse of MatA)

A

From the matrix of minors (from step 2), find the matrix of cofactors
You do this by changing the signs of some of the elements of the matrix of minors according to the rule of alternating signs

45
Q

What is the forth step of finding the inverse of a 3x3 matrix?
(Find the inverse of MatA)

A

Write down the transpose of the matrix of cofactors

46
Q

What is the fith step of finding the inverse of a 3x3 matrix?
(Find the inverse of MatA)

A

The inverse of MatA is given by the formula

47
Q

What are all 5 steps of inverting a 3x3 matrix?

A
48
Q

What can you say about MatA if MatA is equal to the inverse of MatA?

A

MatA is said to be self inverse

49
Q

When MatA is self inverse what does that mean?

A

MatA = Inverse of MatA

50
Q

What is the rule of alternating signs?

A
51
Q

How is a zero matrix denoted?

A

0

52
Q

How is an identity matrix denoted?

A
53
Q

What is a special matrix shortcut to do with inverses?

A

With a MatM -

54
Q

What do you know about

A
55
Q

What may you be asked to prove with Matrices?

A
56
Q

What is another way to prove:

A
57
Q

What do you sometimes forget to do when working out the inverse of a 3x3 matrix?

A

To transpose the matrix of cofactors

zA Level Further Maths (35 decks)

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