Maths › Matrices › Matrix algebra and transformations
Matrix algebra and transformations
Grids of numbers with their own arithmetic, and a second life as movements of the plane: rotations, reflections, enlargements and shears, all stored in four entries.
Builds on Vectors in two dimensions and Graphs, proportion and transformations.
IN THIS TOPIC
- Add and multiply matrices, respecting the size rules and the order of factors.
- Read a transformation matrix from the images of the unit vectors i and j.
- Identify and construct the standard transformations: rotation, reflection, enlargement, shear.
WHAT YOU PROBABLY THINK
Matrix multiplication, like ordinary multiplication, gives the same answer in either order.
An arithmetic for grids
A matrix is a rectangular grid of numbers, added entry by entry when the shapes match. Multiplication is stranger and more useful: the entry in row i, column j of AB comes from row i of A paired with column j of B, products summed. AB exists only when A has as many columns as B has rows.
WORKED EXAMPLE
A product, row by column
For A with rows (2, 1), (0, 3) and B with rows (1, 4), (2, 5), find AB and BA.
AB: top-left is 2 × 1 + 1 × 2 = 4; top-right 2 × 4 + 1 × 5 = 13; bottom row gives 6 and 15. So AB has rows (4, 13), (6, 15).
BA: the same recipe gives rows (2, 13), (4, 17).
AB ≠ BA. Matrix multiplication is not commutative, and the opening claim fails on almost any pair you try; only special pairs commute.
The identity matrix I, ones on the diagonal and zeros elsewhere, multiplies like the number 1: AI = IA = A. It is the do-nothing grid, and it anchors the inverse in the next lesson.
Matrices that move the plane
A 2 × 2 matrix acts on column vectors, and the action is a linear transformation: the origin stays put, straight lines stay straight. The matrix wears its own description openly: its first column is the image of i, its second the image of j. Watch where the two unit vectors go and the matrix writes itself.
The standard catalogue follows from that one fact. Rotation anticlockwise by θ has columns (cos θ, sin θ) and (−sin θ, cos θ). Reflection in y = x swaps the unit vectors: columns (0, 1) and (1, 0). Enlargement by scale factor k about the origin is k times the identity.
YOUR TURN
From picture to matrix
Write down the matrices for reflection in the y-axis, and for the shear that fixes the x-axis and sends (0, 1) to (3, 1).
Show the working
Reflection in the y-axis: i goes to (−1, 0), j stays at (0, 1), so the columns are (−1, 0) and (0, 1).
The shear: i is fixed at (1, 0); j lands on (3, 1). Columns (1, 0) and (3, 1), so the rows read (1, 3) and (0, 1).
In both cases the working is the same two questions: where does i go, where does j go.
Composition is multiplication
Applying transformation B and then transformation A is the single transformation with matrix AB: the right-hand factor acts first, because it sits next to the vector in ABv. Order matters exactly because multiplication is not commutative; a rotation followed by a reflection generally lands somewhere different from the reflection followed by the rotation.
THE EXAM BIT
- Check sizes before multiplying; a product the wrong way round may not even exist.
- Quote the columns-are-images fact when building a matrix; it justifies the answer in one line.
- For combined transformations, write the first-applied matrix on the right.
- Learn the rotation matrix with θ anticlockwise; clockwise questions use −θ.
CHECK YOURSELF
Find the single matrix for reflection in y = x followed by enlargement, scale factor 3, about the origin.
Show a hint
Write the enlargement matrix on the left of the reflection matrix.
Show the answer
T
h
e
r
e
f
l
e
c
t
i
o
n
h
a
s
r
o
w
s
(
0
,
1
)
,
(
1
,
0
)
;
t
h
e
e
n
l
a
r
g
e
m
e
n
t
i
s
3
I
w
i
t
h
r
o
w
s
(
3
,
0
)
,
(
0
,
3
)
.
T
h
e
c
o
m
p
o
s
i
t
e
i
s
(
3
I
)
(
r
e
f
l
e
c
t
i
o
n
)
,
r
o
w
s
(
0
,
3
)
,
(
3
,
0
)
:
e
a
c
h
u
n
i
t
v
e
c
t
o
r
i
s
s
w
a
p
p
e
d
a
n
d
t
h
e
n
t
r
i
p
l
e
d
.
A matrix's columns are the images of i and j; read transformations straight off them.
Doing B then A is the product AB: the matrix applied first sits nearest the vector.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device only.
- Add and multiply matrices, respecting the size rules and the order of factors.
- Read a transformation matrix from the images of the unit vectors i and j.
- Identify and construct the standard transformations: rotation, reflection, enlargement, shear.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.