We begin with a lot of definitions.
A $m\times n$ matrix is a rectangular grid of numbers with $m$ rows and $n$ columns.
A square matrix is one which is $n\times n$ for some $n$.
A (height $m$) column vector is an $m\times 1$ matrix.
A (width $n$) row vector is a $1\times n$ matrix.
${\mathbb{R}}^{n}$ is the set of all column vectors with height $n$ and real numbers as entries, ${\u2102}^{n}$ is the set of all height $n$ column vectors with complex numbers as entries.
${M}_{m\times n}(\mathbb{R})$ is the set of all $m\times n$ matrices with real number entries.
The $m\times n$ zero matrix, written ${0}_{m\times n}$, is the $m\times n$ matrix all of whose entries are zero.
$\left(\begin{array}{c}1\\ 0\end{array}\right)$ is a $2\times 1$ column vector, an element of ${\mathbb{R}}^{2}$.
$\left(\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\end{array}\right)$ is a $2\times 3$ matrix
$\left(\begin{array}{cc}-1& -2\end{array}\right)$ is a $1\times 2$ row vector
$\left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right)$ is a $2\times 2$ square matrix.
${\mathrm{\U0001d7ce}}_{2\times 2}=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$.
The $i,j$ entry of a matrix means the number in row $i$ and column $j$. It is important to get these the correct way round. Usually when you give $(x,y)$ coordinates, $x$ refers to the horizontal direction and $y$ refers to the vertical direction. When we talk about the $i,j$ entry of a matrix, however, the first number $i$ refers to the row number (i.e. the vertical direction) and the second number $j$ refers to the column number (i.e. the horizontal direction).
We often write $A=({a}_{ij})$ to mean that $A$ is the matrix whose $i$, $j$ entry is called ${a}_{ij}$. For example, in the $2\times 2$ case we would have
$$A=\left(\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}& {a}_{22}\end{array}\right).$$ |
If you’re using this notation you must also specify the size of the matrix, of course.
We often talk about the columns and rows of a matrix. If $A$ is an $m\times n$ matrix
$$A=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \mathrm{\cdots}& {a}_{1n}\\ {a}_{21}& {a}_{22}& \mathrm{\cdots}& {a}_{2n}\\ \mathrm{\vdots}& \mathrm{\vdots}& \mathrm{\vdots}& \mathrm{\vdots}\\ {a}_{m1}& {a}_{m2}& \mathrm{\cdots}& {a}_{mn}\end{array}\right)$$ |
then the $i$th row of $A$ means the $1\times n$ row vector
$$\left(\begin{array}{ccc}{a}_{i1}& {a}_{i2}& \mathrm{\cdots}{a}_{in}\end{array}\right)$$ |
and the $j$th column is the $m\times 1$ column vector
$$\left(\begin{array}{c}{a}_{1j}\\ {a}_{2j}\\ \mathrm{\vdots}\\ {a}_{mj}\end{array}\right).$$ |
For example, if
$$A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$$ |
then the first row is $\left(\begin{array}{cc}1& 2\end{array}\right)$ and the second column is $\left(\begin{array}{c}2\\ 4\end{array}\right)$.
We can add matrices of the same size. If $A=({a}_{ij})$ and $B=({b}_{ij})$ are the same size, then $A+B$ is defined to be the matrix whose $i,j$ entry is ${a}_{ij}+{b}_{ij}$.
$$\left(\begin{array}{cc}1& 2\\ 4& 5\end{array}\right)+\left(\begin{array}{cc}0& 1\\ 2& 3\end{array}\right)=\left(\begin{array}{cc}1+0& 2+1\\ 4+2& 5+3\end{array}\right)=\left(\begin{array}{cc}1& 3\\ 6& 8\end{array}\right).$$ |
In other words, we add matrices by adding corresponding entries. We never add matrices of different sizes.
We also multiply matrices by numbers. This is called scalar multiplication. If $A=({a}_{ij})$ is a matrix and $\lambda $ a number then $\lambda A$ means the matrix obtained by multiplying every entry in $A$ by $\lambda $, so the $i,j$ entry of $\lambda A$ is $\lambda {a}_{ij}$.
$$2\left(\begin{array}{cc}1& -3\\ 0& 1\end{array}\right)=\left(\begin{array}{cc}2& -6\\ 0& 2\end{array}\right).$$ |
These operations have some familiar properties.
If $a$ and $b$ are numbers and $A$, $B$, and $C$ are matrices of the same size,
$A+B=B+A$ (commutativity)
$A+(B+C)=(A+B)+C$ (associativity)
$(a+b)A=aA+bA$ (distributivity),
$a(A+B)=aA+aB$ (distributivity), and
$a(bA)=(ab)A$. ∎
These can be proved using the usual laws for addition and multiplication of numbers.