Matrices

Question 1

When given the dimensions of a matrix what order do you get given it e.g a 2x3 matrix?

Answer 1

Rows x Columns

Question 2

What is the case when adding or subtracting matrices?

Answer 2

You add each term (top left + top left or top right + top right)
Note to add matrices they must have exactly the same dimensions

Question 3

How do you multiply a matrix by a scalar?

Answer 3

Multiply every element in the matrix by the scalar

Question 4

How do you multiply matrices?

Answer 4

(top left * top left + top right * bottom left top left * top right + top right * bottom right)
(bottom left * top left + bottom right * bottom left bottom left * top right + bottom right * bottom right)
First term of first matrix * first term of second matric then go across a column in first matrix and down a row in second matrix
First row in first matrix relates to first row in second matrix
Second row in first matrix relates to second row in second matrix

Note to multiply matrices they must have the same number of rows

Question 5

What is an identity matrix?

Answer 5

I is the identity matrix where the values in the leading diagonal are all 1 and every other value is 0

Note the leading diagonal is top left to bottom right and the matrix must be square

Question 6

How do you find the determinant of a 2x2 matrix?

Answer 6

top left * bottom right - top right * bottom left = det

Question 7

How do you multiply modulus of matrices?

Answer 7

|a|*|b| = |ab|

Question 8

How do you find the inverse of a 2x2 matrix?

Answer 8

Swap the entries in the leading diagonal and make the other entries negative. Then divide the original matrix by its determinant

Question 9

WHat are some things used to solve matrix equations?

Answer 9

MM^-1 = M^-1M = I
(M^-1)^-1 = M
(AB)^-1 = B^-1A^-1

Question 10

How do you pre multiply to rearrange AB=C for B?

Answer 10

MUltiply both sides (on the furthest left) by A^-1

Question 11

What is the case if a matrix is singular?

Answer 11

det(M) = 0 and no inverse matrix exists

Question 12

How do you find the determinant of a 3x3 matrix?

Answer 12

Determinant of the matrix by removing the row and column containing the top left element - the same for the next element in the row + the same for the next element in the row

Question 13

How do you find the inverse of a 3x3 matrix?

Answer 13

• Find the determinant of the matrix
• Find the minor of each element (replace each term with the determinant of the matrix you get from removing the row and column that contain that term)
• Apply the matrix of signs, everything except from the top right to bottom left and top left to bottom right diagonals gets multiplied by -1
• Transpose the matrix (first row becomes first column second row becomes second column and third row becomes third column)
• Divide by the original matrix’s determinant

Question 14

How can simultaneous equations be represented in matrix form?

Answer 14

(matrix of coefficients (2x2 or 3x3)) * (matrix of variables (1x2 or 1x3)) = constants equations are equal too

The equations literaly just go into each row (term term = constant)

Question 15

For simultaneous equation matrices if the coefficient matrix has a det of 0 then what is the case?

Answer 15

The equations have 0 or infinite solutions

If the det is not 0 then the equations have a unique solution

Question 16

How are these cases represented for a couple of simultaneous equations?
Parallel lines:
Same line:
Not parallel not the same line:

Answer 16

Parallel lines: no solutions
Same line: infinite solutions
Not parallel not the same line: unique solution

Question 17

How are these cases represented geometrically for a trio of simultaneous equations?
A unique solution:
Infinitely many solutions:
No solutions:

Answer 17

A unique solution: Three planes intersect at one point
Infinitely many solutions: The planes form a sheaf (common line) or two or three planes are the same
No solutions: Two or three planes are parallel or the three planes form a prism

Question 18

Modelling…

Question 19

What kind of matrix represents a linear transformation?

Answer 19

A 2x2 matrix

Question 20

What are the four possible types of linear transformations?

Answer 20

Reflection by a line through the origin
Rotation about (0,0) through any angle
Stretch parallel to x or y axis
Enlargement about (0,0) SF k

Question 21

How do you find what a linear tranformation does given its matrix?

Answer 21

Matrix * Unit square coordinates = New coordinate

Draw the new and old coordinates of the unit square on a grid and see what transformation gets you from the unit square to your new shape
Note the unit square just has coordinates ( (0,0) (1,0) (0,1) (1,1) )

Question 22

How do you combine multiple linear transformations e.g A followed by B?

Answer 22

matrix B * matrix A
BA
If a matrix transforms a point then the inverse matrix returns the point back to its original

Question 23

What does the determinant of a 2x2 matrix represent geometrically?

Answer 23

The area SF of the transformation

If the det is negative this represents that a reflection has taken place

Question 24

What is an invariant point?

Answer 24

A point that remains unchanged under a transformation (a point that maps onto its self)

Question 25

What is a line of invariant points?

Answer 25

A line where every point on the line is mapped onto its self

Question 26

How can you show that every point on a line is invariant (that the line is a line of invariant points) under a transformation? Line y=ax

Answer 26

matrix * (x ax) = (x ax)
and prove this is always true

Question 27

How do you find the invariant points under a transformation?

Answer 27

matrix * (x y) = (x y)
Solve simultaneous equations and there solutions can be interpreted

Question 28

What is an invariant line?

Answer 28

A line which under a transformation every point maps not neccesarily onto its self but onto another point on the line

Question 29

How do you find all the invariant lines under a transformation? The general line equation is mx+c

Answer 29

• matrix * (x mx+c) = …
• sub … into y=mx+c for y and x
• Set this equation = 0 and factorise out x to get an equation in terms of m and set tis equal too 0 and the equation with c in must also be equal too 0
• The quadratic solves for m and c is solved for using intuition generally. These values of m and c give invariant lines

Question 30

What two types of 3d transformations do you need to know?

Answer 30

Reflections in x,y or z plane
Rotations about x,y or z axis

x=0 is yz plane
y=0 is xz plane
z=0 is xy plane

about x axis - towards positive z
about y axis - towards positive x
about z axis - towards positive y

Question 31

How do you find the rotation matrices for 3d matrices?

Answer 31

Rotation anticlock wise about x axis - first column and first row of 3x3 matrix get swapped with (1 0 0)
Rotation anticlockwise about y axis - second column and second row get swapped with (0 1 0)
Rotation anticlockwise about z axis - third row and third column get swapped with (0 0 1)

The rest of the matrix in each case is filled in by the 2x2 rotaion matrix that is in the formula book