Matrices

Question 1

How are matrices added/subtracted?

Answer 1

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

Question 2

How are matrices multiplied?

Answer 2

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.

Question 3

How can you divide by a matrix?

Answer 3

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

Question 4

How do you find the inverse of a matrix?

Answer 4

͟ ͟ ͟1͟ ͟ _ ( d -b )

ad-bc ( -c a )

Question 5

What is the determinant?

Answer 5

ad-bc

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

Question 6

What is a singular matrix?

Answer 6

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

Question 7

What is the identity matrix?

Answer 7

( 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

Question 8

What is the zero matrix?

Answer 8

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

Question 9

What is the rotation matrix?

Answer 9

For a rotation of θ° anticlockwise from the origin:

cosθ -sinθ
( sinθ cosθ )

Question 10

What is the reflection matrix?

Answer 10

For a reflection in the line y = tanθ x :

cos2θ sin2θ
( sin2θ -cos2θ )

Question 11

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

Answer 11

( -1 0 )

0 1

Question 12

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

Answer 12

( 1 0 )

0 -1

Question 13

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

Answer 13

( 0 1 )

1 0

Question 14

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

Answer 14

( 0 -1 )

-1 0

Question 15

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

Answer 15

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

Question 16

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

Answer 16

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

Question 17

What is the matrix for an enlargement from the origin?

Answer 17

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

Question 18

How do you find the transformation matrix?

Answer 18

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

Question 19

How do you find invariant points?

Answer 19

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.

Question 20

How do you find invariant lines?

Answer 20

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