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AQA A-Level Further Maths > Matrices > Flashcards

Flashcards in Matrices Deck (20)
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1

How are matrices added/subtracted?

Add/subtract the elements in the same positions (the matrices must be the same size)

2

How are matrices multiplied?

Scalar multiplication:
Multiply each element of the matrix by the scalar.

Matrix multiplication:
Multiply the first element in the first row of the first martix by the first element in the first column of the second matrix. Do so for all the elements in the first row of the first matrix and the first column of the second matrix, and add the products together. Repeat for each row of the first matrix and each column of the scond matrix.

3

How can you divide by a matrix?

Find the inverse of the matrix and multiply by the inverse.

4

How do you find the inverse of a matrix?

͟ ͟ ͟1͟ ͟ _ ( d -b )
ad-bc ( -c a )

5

What is the determinant?

ad-bc

If the determinant is 0, the matrix must be singular (because you can't divide by 0).

6

What is a singular matrix?

A matrix that does not have an inverse - this is when the determinant = 0 (can't divide by 0)

7

What is the identity matrix?

( 1 0 )
( 0 1 )
Denoted as I
All elements of the matrix are 0, apart from those in the lead diagonal which are 1
For matrix A : AI = A

8

What is the zero matrix?

( 0 0 )
( 0 0 )
Denoted as 0
All elements are 0
For matrix A : A0 = 0

9

What is the rotation matrix?

For a rotation of θ° anticlockwise from the origin:

( cosθ -sinθ )
( sinθ cosθ )

10

What is the reflection matrix?

For a reflection in the line y = tanθ x :

( cos2θ sin2θ )
( sin2θ -cos2θ )

11

What is the matrix for a reflection in the y axis?

( -1 0 )
( 0 1 )

12

What is the matrix for a reflection in the x axis?

( 1 0 )
( 0 -1 )

13

What is the matrix for a reflection in the line y = x?

( 0 1 )
( 1 0 )

14

What is the matrix for a reflection in the line y = -x?

( 0 -1 )
( -1 0 )

15

What is the matrix for a stretch in the x axis?

For scale factor k :
( k 0 )
( 0 1 )

16

What is the matrix for a stretch in the y axis?

For scale factor k :
( 1 0 )
( 0 k )

17

What is the matrix for an enlargement from the origin?

For scale factor k :
( k 0 )
( 0 k )

18

How do you find the transformation matrix?

Use vectors with a 1 and the rest 0 (so that if merged, would make the identity matrix) and see where they transform to
Merge the new vectors to form a matrix

19

How do you find invariant points?

Invariant points are points that don't move after a transformation.
Transform x and y under the transformation matrix.
Equate these to x and y respectively and form equations.
Solve the equations simultaneously.

20

How do you find invariant lines?

Invariant lines are lines whose points map onto the line after a transformation.
Transform x and y under the transformation matrix.
Equate these to x and y respectively and form equations.
Substitute mx for y and solve for m.
The answer is y = mx