Further Maths - Matrices
See also:
- Further Maths - DeterminantsA
- Further Maths - Inverting MatricesA
- Further Maths - Solving Systems of Equations Using MatricesA
- Further Maths - Linear TransformationsA
Dimensions
What are the dimensions of a matrix?
What are the dimensions of
\[\begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \\ 3 & 3 & 3 \end{matrix}\]
?
What are the dimensions of
\[\begin{matrix} 3 & 3 & 3 \\ 3 & 3 & 3 \end{matrix}\]
?
What are the dimensions of
\[\begin{matrix} 3 & 3 \\ 3 & 3 \\ 3 & 3 \end{matrix}\]
?
Visualise a
\[5 \times 2\]
matrix?
Adding and multiplying
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?
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?
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 $4 \times 9$?
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.
Coordinate matrices
What’s the co-ordinate matrix for
\[A(2,1), B(2,7), C(5,1)\]
?
What are the co-ordinates
\[A, B\]
and
\[C\]
for
\[\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix}\]
?
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.
Transformation matrices
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 $\hat{j}$.
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.
How is matrix multiplication different from normal multiplication?
It is not commutative.
What’s a way of describing something not being commutative?
The identity and inverse
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?
What is the
\[3 \times 3\]
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.
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.
Conformability
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.
Determinants
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 $4$.
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.