See also:
- [[Further Maths - Determinants]]S
- [[Further Maths - Inverting Matricies]]S
- [[Further Maths - Solving Systems of Equations Using Matricies]]S
- [[Further Maths - Linear Transformations]]S
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 matricies??
When they have the same dimensions.
When can you multiply matricies??
The columns of the first matrix equals the rows of the second matrix.
If two matricies 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 matricies 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 matricies??
Multiplying all the transformation matricies 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 matricies 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 matricies 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 matricies??
The inverse matrix of $A$.
How can you sort of define division for matricies??
Inversion, finding the matrix you can multiply both sides by to eliminate one of the matricies.
What does it mean for two matricies to be multiplicatively conformable??
They can be multiplied.
The fancy term for two matricies that can be multiplied is??
Multiplicatively conformable.
What does it mean for two matricies to be additively conformable??
They can be added.
The fancy term for two matricies 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 $4$.
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date: 2020-09-23 19:02
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title: Further Maths - Matricies