Perform matrix operations and analyse graphs, paths and networks

What this dot point is asking

This dot point joins two connected ideas: matrices as a way to store and combine numbers in a grid, and networks as diagrams of connected objects that can themselves be written as matrices.

Matrix operations

A matrix has order $m \times n$ (rows by columns). You can add or subtract two matrices only if they have the same order, by combining matching entries.

Matrix multiplication of $A$ ($m \times n$) by $B$ ($n \times p$) is defined only when the number of columns of $A$ equals the number of rows of $B$. The result is $m \times p$, and each entry is the dot product of a row of $A$ with a column of $B$.

Determinant and inverse of a 2x2 matrix

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is $\det(A) = ad - bc$. The inverse exists only when $\det(A) \neq 0$:

The inverse lets you solve a matrix equation $AX = B$ by computing $X = A^{-1}B$.

Graphs and networks

A graph (network) is a set of vertices (nodes) joined by edges. Key terms: the degree of a vertex is the number of edges meeting it; a path is a walk that does not repeat vertices; a connected graph has a route between every pair of vertices.

An adjacency matrix records how many edges directly join each pair of vertices. For an undirected graph this matrix is symmetric.

Shortest path

In a weighted network each edge carries a number (distance, cost or time). The shortest path between two vertices is the route with the smallest total weight. You find it by systematically labelling each vertex with the least cumulative distance from the start, then reading the smallest total at the destination and tracing back.

In the exam, watch the order of matrix multiplication, confirm the determinant is non-zero before inverting, and use matrix powers to count walks rather than tracing every path by hand.