# Further Maths - Matrices

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/matrices/ · Updated: 2026-06-05 · Tags: further-maths, latex-block-alt, school

See also:

- [Further Maths - Determinants](https://ollybritton.com/notes/a-level/further-maths/topics/determinants/)
- [Further Maths - Inverting Matrices](https://ollybritton.com/notes/a-level/further-maths/topics/inverting-matrices/)
- [Further Maths - Solving Systems of Equations Using Matrices](https://ollybritton.com/notes/a-level/further-maths/topics/solving-systems-of-equations-matrices/)
- [Further Maths - Linear Transformations](https://ollybritton.com/notes/a-level/further-maths/topics/linear-transformations/)

### Dimensions
##### If a matrix is $m \times n$, how many columns does it have??
$n$ columns.
^columns-of-m-by-n-matrix

##### If a matrix is $m \times n$, how many rows does it have??
$m$ rows.
^rows-of-m-by-n-matrix

##### What are the dimensions of a matrix??
$$
\text{rows} \times \text{columns}
$$
^matrix-dimensions-convention

##### What are the dimensions of $$\begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \end{matrix}$$??
$$
3 \times 3
$$
^matrix-dimensions-square-example

##### What are the dimensions of $$\begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \end{matrix}$$??
$$
2 \times 3
$$
^matrix-dimensions-wide-example

##### What are the dimensions of $$\begin{matrix} 3 & 3 \\ 3 & 3 \\ 3 & 3 \end{matrix}$$??
$$
3 \times 2
$$
^matrix-dimensions-tall-example

##### Visualise a $$5 \times 2$$ matrix??
$$
\begin{matrix} 5 & 5 \\ 5 & 5 \\ 5 & 5 \\ 5 & 5 \\ 5 & 5 \end{matrix}
$$
^visualise-5-by-2-matrix

### Adding and multiplying
##### When can you add matrices??
When they have the same dimensions.
^matrix-addition-condition

##### When can you multiply matrices??
The columns of the first matrix equals the rows of the second matrix.
^matrix-multiplication-condition

##### If two matrices are $m_1 \times n_1$ and $m_2 \times n_2$, how can you tell whether you can multiply them??
$$
n_1 = m_2
$$
^matrix-multiplication-dimension-test

##### If two matrices are $m_1 \times n_1$ and $m_2 \times n_2$, what will be the size of the resulting multiplied matrix??
$$
m_1 \times n_2
$$
^matrix-product-dimensions

##### Can you multiply a $5 \times 2$ and a $3 \times 4$ matrix??
No.
^matrix-multiplication-example-incompatible

##### Can you multiply a $6 \times 3$ and a $3 \times 4$ matrix??
Yes.
^matrix-multiplication-example-compatible

##### What will be the size of the multiplied matrix if you multiply $3 \times 4$ and $4 \times 9$??
$$
3 \times 9
$$
^matrix-product-dimensions-example

##### What's a one sentence explanation for matrix multiplication??
You multiply all the rows of the first matrix by the columns of the second matrix.
^matrix-multiplication-in-words

### Coordinate matrices
##### What's the co-ordinate matrix for $$A(2,1), B(2,7), C(5,1)$$??
$$
\begin{matrix} 2 & 2 & 5 \\ 1 & 7 & 1 \end{matrix}
$$
^coordinate-matrix-from-points

##### What are the co-ordinates $$A, B$$ and $$C$$ for $$\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}$$??
$$
A(1,4) \\
B(2,5) \\
C(3,6)
$$
^coordinates-from-matrix

##### What is a co-ordinate matrix??
A way of representing co-ordinates in a matrix, with all the co-ordinates top-down next to each other.
^coordinate-matrix-definition

### Transformation matrices
##### What is a transformation matrix??
A matrix which describes a transformation to a coordinate system.
^transformation-matrix-definition

##### How can you combine multiple transformation matrices??
Multiplying all the transformation matrices together.
^combining-transformation-matrices

##### How can you think of a transformation matrix??
As defining new, transformed values for the unit vectors $\hat{i}$ and $\hat{j}$.
^transformation-matrix-as-unit-vectors

### Properties of multiplication
##### The rule for matrices that $(A \times B) \times C = A \times (B \times C)$ is known as??
The Law of Associativity.
^matrix-multiplication-associativity

##### How is matrix multiplication different from normal multiplication??
It is not commutative.
^matrix-multiplication-non-commutative

##### What's a way of describing something not being commutative??
$$
AB \neq BA
$$
^matrix-multiplication-non-commutative-notation

### The identity and inverse
##### What is the identity matrix??
The $m \times n$ matrix which does not change what it is multiplying.
^identity-matrix-definition

##### What is the $$2 \times 2$$ identity matrix??
$$
\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}
$$
^identity-matrix-2-by-2

##### What is the $$3 \times 3$$ identity matrix??
$$
\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}
$$
^identity-matrix-3-by-3

##### $I$ in matrices means...??
The identity matrix.
^identity-matrix-symbol

##### What is the inverse matrix of a matrix??
A matrix that multiplies with the original matrix to give the $m \times n$ identity matrix.
^inverse-matrix-definition

##### What does $A^{-1}$ mean in matrices??
The inverse matrix of $A$.
^inverse-matrix-notation

##### How can you sort of define division for matrices??
Inversion, finding the matrix you can multiply both sides by to eliminate one of the matrices.
^matrix-division-as-inversion

### Conformability
##### What does it mean for two matrices to be multiplicatively conformable??
They can be multiplied.
^multiplicatively-conformable-meaning

##### The fancy term for two matrices that can be multiplied is??
Multiplicatively conformable.
^multiplicatively-conformable-term

##### What does it mean for two matrices to be additively conformable??
They can be added.
^additively-conformable-meaning

##### The fancy term for two matrices that can be added is??
Additively conformable.
^additively-conformable-term

### Determinants
##### How can you think of the determinant of a transformation matrix??
The scale factor.
^determinant-of-transformation-as-scale-factor

##### If a $2 \times 2$ matrix has a determinant of $4$, then what does that tell you about the transformation??
It increases the area by a factor of $4$.
^determinant-area-scale-factor

##### If a $3 \times 3$ matrix has a determinant of $4$, then what does that tell you about the transformation??
It increases the volume by a factor of $4$.
^determinant-volume-scale-factor

### A shortcut for the inverse
##### If you've been given that $$\pmb{M}\pmb{M}^T = 4\pmb{I}$$ how can you quickly work out the inverse matrix??
$\pmb{M}^T$ must be equal to $4$ times the inverse matrix.
^inverse-from-orthogonal-scaling

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