Addition, subtraction and scalar multiplication
Two matrices can be added or subtracted only when they have the same order; we simply combine corresponding elements. To multiply a matrix by a scalar $k$, multiply every element by $k$.
Example
$\begin{pmatrix}1&2\\3&4\end{pmatrix}+\begin{pmatrix}5&0\\-1&2\end{pmatrix}=\begin{pmatrix}6&2\\2&6\end{pmatrix}$, and $3\begin{pmatrix}2&-1\\0&4\end{pmatrix}=\begin{pmatrix}6&-3\\0&12\end{pmatrix}$.
Matrix multiplication
The product $AB$ exists only when the number of columns of $A$ equals the number of rows of $B$. Each entry is the sum of products of a row of $A$ with a column of $B$. Order matters: usually $AB\ne BA$. A $(2\times 3)$ times a $(3\times 4)$ gives a $(2\times 4)$ matrix.
Inverse of a 2×2 matrix
For $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, the determinant is $\det A=ad-bc$. If $\det A\ne 0$, the inverse is given below. When $\det A=0$ the matrix has no inverse.
Key formula
$A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$
Solving simultaneous equations
Equations such as $\begin{cases}2x+y=5\\3x+2y=8\end{cases}$ can be written as $\begin{pmatrix}2&1\\3&2\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}5\\8\end{pmatrix}$. Then $\begin{pmatrix}x\\y\end{pmatrix}=A^{-1}\begin{pmatrix}5\\8\end{pmatrix}$, giving $x=2,\ y=1$.
Remember
Swap the diagonal ($a\leftrightarrow d$), negate the off-diagonal ($b,c$), then divide by the determinant.