# Further Maths - Matricies

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/matricies/ · Updated: 2020-09-23 · 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 Matricies](https://ollybritton.com/notes/a-level/further-maths/topics/inverting-matricies/)
- [Further Maths - Solving Systems of Equations Using Matricies](https://ollybritton.com/notes/a-level/further-maths/topics/solving-systems-of-equations-matricies/)
- [Further Maths - Linear Transformations](https://ollybritton.com/notes/a-level/further-maths/topics/linear-transformations/)

##### If a matrix is $m \times n$, how many columns does it have??
$m$ columns.

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

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

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

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

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

##### Visualise a $$5 \times 2$$ matrix??
$$
\begin{matrix} 5 & 5 \\ 5 & 5 \\ 5 & 5 \\ 5 & 5 \\ 5 & 5 \end{matrix}
$$

##### When can you add matrices??
When they have the same dimensions.

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

##### If two matrices are $m_1 \times n_1$ and $m_2 \times n_2$, how can you tell whether you can multiply them??
$$
n_1 = m_2
$$

##### 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??
$$
m_1 \times n_2
$$

##### Can you multiply a $5 \times 2$ and a $3 \times 4$ matrix??
No.

##### Can you multiply a $6 \times 3$ and a $3 \times 4$ matrix??
Yes.

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

##### 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.

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

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

##### 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.

##### What is a transformation matrix??
A matrix which describes a transformation to a coordinate system.

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

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

##### The rule for matrices that $(A \times B) \times C = A \times (B \times C)$ is known as??
The Law of Associativity.

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

##### What's a way of describing something not being commutative??
$$
AB \neq BA
$$

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

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

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

##### $I$ in matrices means...??
The identity matrix.

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

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

##### 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.

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

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

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

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

##### How can you think of the determinant of a transformation matrix??
The 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 $4$.

##### 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 $4$.

### 2022-04-14
##### 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.

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt