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Academic Skills Advice
Matrices Summary
To transpose a matrix write the rows as columns.
For example:
2 1 5
A = [1 5 0
2 1 4
0
9]
2
2
1
5
0
AT =
1
5
0
9
2
1
4
2
To add or subtract matrices they must be the same dimensions. Just add or subtract the
corresponding numbers.
For example:
2 5 −1
[4 1 2] +
−3 3 4
3
[7
10
−2 3
5
4
2 ] = [11
5 −5
7
3 2
5 4]
8 −1
To multiply matrices the number of columns in the 1st must equal the number of rows in
the 2nd because you multiply across the row and down the column.
For example:
(6 + 10) (−9 + 14)
3 2
2 −3 7
[4 6 ] × [
] = [(8 + 30) (−12 + 42)
5 7 8
(2 − 25) (−3 − 35)
1 −5
16
= [ 38
−23
(21 + 16)
(28 + 48)]
(7 − 40)
5
37
30
76 ]
−38 −33
Transformations
When setting up the multiplication for a transformation the transformation matrix always
goes first.
Rotations:
Clockwise: [
𝑐𝑜𝑠𝜃
−𝑠𝑖𝑛𝜃
𝑠𝑖𝑛𝜃
]
𝑐𝑜𝑠𝜃
Reflections:
0
[
1
1
] : swaps 𝑥 𝑎𝑛𝑑 𝑦.
0
𝑐𝑜𝑠𝜃
Anticlockwise: [
𝑠𝑖𝑛𝜃
−𝑠𝑖𝑛𝜃
]
𝑐𝑜𝑠𝜃
Scaling / magnification:
[
3 0
] : makes 𝑥 3x bigger and 𝑦 2x bigger
0 2
Composite:
If a matrix A is reflected by T1 then magnified by T2 to give matrix B then the single
transformation that would transform A into B is T 2T1 (in that order).
H Jackson 2010 / Academic Skills
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The Inverse (A-1):
AA = I
-1
1
(the identity matrix) e.g. [0
0
0 0
1 0]
0 1
or [
1 0
]
0 1
Finding the Inverse:
1
𝑎
If: A = [
𝑐
A-1 = |𝐴| x adj
𝑏
]
𝑑
|𝐴| = (𝑎𝑑) − (𝑏𝑐)
Where: |𝐴| = determinant
adj = adjoint
adj = [
𝑑
−𝑐
−𝑏
]
𝑎
n.b. If det=0 matrix is singular and there is no inverse. If det≠0, matrix is non-singular.
Solving Simultaneous Equations :
If :
then:
𝐴𝑥 = 𝐵
𝑥 = 𝐴−1 𝐵
where :
𝐴 is the matrix
𝑥 is the 𝑥, 𝑦, 𝑧 (𝑒𝑡𝑐) values
𝐵 is the answers
All you need to remember to find the values of 𝑥, 𝑦, 𝑧 etc is to do: inverse x answers
Example:
We have:
5𝑥 − 6𝑦 = −8
−3𝑥 + 4𝑦 = 6
Solve
and
[
−8
5 −6 𝑥
] [𝑦] = [ ]
6
−3 4
𝐴
So:
simultaneously
𝑥 =
𝐵
𝑥
−8
[𝑦] = 𝐴−1 [ ]
6
𝑥
[𝑦 ] =
H Jackson 2010 / Academic Skills
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[
2 3
1
1 4
6 −8
] [ ]= [ ]
2 6
5 6
(Use the method shown above to find the
inverse then multiply by the answers.)
=
2
[ ]
3
2