=== Examples ===

  * $\begin{bmatrix}3\\2\\4\\0\\-1\end{bmatrix}$ is a $5\times 1$ matrix. A matrix like this with one column is called a **column vector**.
  * $\begin{bmatrix}3&2&4&0&-1\end{bmatrix}$ is a $1\times 5$ matrix. A matrix like this with one row is called a **row vector**.

Even though the row matrix and the column matrix above have the same entries, they have a different "shape", or "size", so we must think of them has being different matrices. Let's give the definitions to make this precise.

=== Definition ===

{{page>same size}}

=== Definition ===

{{page>equal matrices}}

=== Examples ===

  * $\begin{bmatrix}3\\2\\4\\0\\-1\end{bmatrix}\ne \begin{bmatrix}3&2&4&0&-1\end{bmatrix}$, since these matrices have different sizes: the first is $5\times 1$ but the second is $1\times 5$.
==== ====
  * $\begin{bmatrix}1\\2\end{bmatrix}\ne\begin{bmatrix}1 &0\\2&0\end{bmatrix}$ since these matrices are not the same size.
  * $\begin{bmatrix}1&0\\0&1\end{bmatrix}\ne \begin{bmatrix}1&0\\1&0\end{bmatrix}$ because even though they have the same size, the $(2,1)$ entries are different.
  * If $\begin{bmatrix}3x&7y+2\\8z-3&w^2\end{bmatrix}=\begin{bmatrix}1&2z\\\sqrt2&9\end{bmatrix}$ then we know that all the corresponding entries are equal, so we get four equations:\begin{align*}3x&=1\\7y+2&=2z\\8z-3&=\sqrt2\\w^2&=9\end{align*}

===== Operations on matrices =====

We want to define operations on matrices: some (useful) ways of taking two matrices and making a new matrix.

Before we begin, a remark about $1\times 1$ matrices. These are of the form $[a_{11}]$ where $a_{11}$ is just a number. The square brackets $[\,]$ don't really matter here; they just keep the inside of a matrix in one place. So really: a $1\times 1$ matrix is just a number. This means that special cases of the operations we define will be operations on ordinary numbers. You should check that in the special case when all the matrices involved are $1\times 1$ matrices, the operations become the ordinary operations on numbers, so we are //generalising// the familiar operations (addition, subtraction, multiplication and so on) from numbers to matrices.

==== Matrix addition and subtraction ====

=== Definition of matrix addition ===

{{page>matrix addition}}

=== Example ===
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}+\begin{bmatrix}-2&2&0\\1&1&1\end{bmatrix}=\begin{bmatrix}-1&4&-2\\4&1&6\end{bmatrix}.\]

=== Example ===
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}+\begin{bmatrix}-2&2\\1&1\end{bmatrix}\text{ is undefined.}\]

=== Remarks ===

  - For any matrices $A$ and $B$ with the same size: $A+B=B+A$. We say that matrix addition is //commutative//.
  - For any matrices $A$, $B$ and $C$ with the same size: $(A+B)+C=A+(B+C)$. We say that matrix addition is //associative//.

=== Definition of the zero matrix ===

{{page>zero matrix}}

=== Exercise ===

Show that if $A$ is any $n\times m$ matrix, then 
\[ 0_{n\times m}+A=A=A+0_{n\times m}.\]
Remember that when checking that matrices are equal, you have to check that they have the same size, and that all the entries are the same.


=== Definition of matrix subtraction ===

{{page>matrix subtraction}}

=== Example ===
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}-\begin{bmatrix}-2&2&0\\1&1&1\end{bmatrix}=\begin{bmatrix}3&0&-2\\2&-1&4\end{bmatrix}.\]

=== Example ===
\[ \begin{bmatrix}1&2&-2\\3&0&5\end{bmatrix}-\begin{bmatrix}-2&2\\1&1\end{bmatrix}\text{ is undefined.}\]



==== Scalar multiplication ====

=== Definition of a scalar ===

{{page>scalar}}

=== Definition of scalar multiplication of matrices ===

{{page>scalar multiplication of matrices}}

=== Example ===
If $A=\begin{bmatrix}1&0&-3\\3&-4&1\end{bmatrix}$, then $3A=\begin{bmatrix}3&0&-9\\9&-12&3\end{bmatrix}$. In other words,
\[ 3\begin{bmatrix}1&0&-3\\3&-4&1\end{bmatrix}=\begin{bmatrix}3&0&-9\\9&-12&3\end{bmatrix}.\]

=== The negative of a matrix ===
{{page>matrix negation}}

=== Exercise ===
Prove that $A-B=A+(-B)$ for any matrices $A$ and $B$ of the same size.

==== Row-column multiplication ====

=== Definition of row-column multiplication ===
{{page>row-column multiplication}}

=== Examples ===

  * $\begin{bmatrix}1&2\end{bmatrix}\begin{bmatrix}3\\-1\end{bmatrix}=1\cdot 3+2\cdot(-1)=3+(-2)=1$.
  * $\begin{bmatrix}1&2&7\end{bmatrix}\begin{bmatrix}3\\-1\end{bmatrix}$ is not defined.
  * $\begin{bmatrix}2&3&5\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=2x+3y+5z$.
  * Generalising the previous example: if $a=\begin{bmatrix}a_1&a_2&\dots&a_m\end{bmatrix}$ and $x=\begin{bmatrix}x_1\\x_2\\\vdots\\x_m\end{bmatrix}$, then $ax=a_1x_1+a_2x_2+\dots+a_mx_m$. So we can write any [[linear equation]] $a_1x_1+a_2x_2+\dots+a_mx_m=b$ as a shorter matrix equation: $ax=b$.

==== Matrix multiplication ====

This generalises row-column multiplication. The idea is that you build a new matrix from all possible row-column products. The formal definition will appear later, but here's an example:
\[ \def\r{\begin{bmatrix}1&0&5\end{bmatrix}}\def\rr{\begin{bmatrix}2&-1&3\end{bmatrix}}\begin{bmatrix}1&0&5\\2&-1&3\end{bmatrix}\begin{bmatrix} 1&2\\3&4\\5&6\end{bmatrix}
\def\s{\begin{bmatrix}1\\3\\5\end{bmatrix}}\def\ss{\begin{bmatrix}2\\4\\6\end{bmatrix}}
=
\begin{bmatrix}{\r\s}&{\r\ss}\\{\rr\s}&{\rr\ss}\end{bmatrix}=\begin{bmatrix}26&32\\14&18\end{bmatrix}.\]