Unit 4 Matrices

13 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

How does a permutation matrix rearrange the rows of a state matrix, and how are binary matrices used to encode and reorder information?

Recognise and use a permutation matrix as a binary matrix with exactly one 1 in each row and column, apply it to reorder the entries of a state matrix, and identify the matrix that reverses or repeats the reordering

The VCE General Mathematics Unit 4 binary and permutation matrices: telling a binary matrix from a permutation matrix, reordering a column matrix row by row, reading a movement matrix column by column, the transpose as the inverse, and powers that repeat a rotation, with the real 2023 and 2025 VCAA questions.

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What is a tree and a spanning tree, and how are minimum spanning trees found by inspection or by Prim's algorithm to solve minimal connector problems?

Identify trees and spanning trees, determine a minimum spanning tree of a weighted connected graph by inspection or by Prim's algorithm (from a network or a table), and use minimum spanning trees to solve minimal connector problems

The VCE General Mathematics Unit 4 minimal connector topic. Properties of trees (n vertices, n minus 1 edges, no cycles), spanning trees, finding a minimum spanning tree by inspection or with Prim's algorithm from a diagram or a table, forced edges and unknown weights, and how it differs from a shortest path, with real 2023 and 2025 VCAA questions.

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How do transition diagrams and transition matrices model step-by-step change in a system, and how are future, past and equilibrium states found, including with culling and restocking?

Construct a transition matrix from a transition diagram or written description, use the matrix recurrence relation S(n+1) = TS(n) to generate a sequence of state matrices, informally identify the equilibrium state of a regular transition matrix, and use S(n+1) = TS(n) + B to model populations with culling and restocking

The VCE General Mathematics Unit 4 transition matrix topic. Reading and building transition diagrams and matrices (columns are this step, rows are next step), stepping state matrices forward and backward, counting how many move between states, regular matrices and the equilibrium state, absorbing states, and the culling and restocking recurrence, with real 2023 and 2025 VCAA questions.

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What is the difference between Eulerian and Hamiltonian travel, and how do vertex degrees decide whether an Eulerian trail or circuit exists?

Distinguish walks, trails, paths, cycles, Eulerian trails and circuits, and Hamiltonian paths and cycles, and use the number of odd-degree vertices to decide whether an Eulerian trail or circuit exists in a connected graph

The VCE General Mathematics Unit 4 exploring and travelling problems: walks, trails, paths, cycles and circuits, the odd-degree test for Eulerian trails and circuits, which edges to repeat for a shortest inspection route, and Hamiltonian paths and cycles, with the real 2023 and 2025 VCAA questions.

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How do you find the greatest flow a network can carry from source to sink, and how does the minimum cut prove the maximum flow?

Model a directed capacitated network with a source and sink, find the maximum flow from source to sink, identify cuts and their capacities, and use the minimum cut to confirm the maximum flow

The VCE General Mathematics Unit 4 flow problems: capacity, source and sink, the bottleneck on a path, finding a maximum flow by inspection, what counts as a cut, counting only edges from the source side to the sink side, the maximum-flow minimum-cut theorem, and which change increases the flow, with the real 2023 and 2025 VCAA questions.

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How is a project scheduled with critical path analysis, including dummy activities, float and crashing, and how does this connect to flow and allocation problems?

Construct an activity network from a precedence table, including dummy activities where necessary, use forward and backward scanning to find earliest and latest start times, identify the critical path and float times, and use crashing to reduce the completion time of a project

The VCE General Mathematics Unit 4 scheduling topic. Building an activity network from a precedence table, when and where a dummy activity is needed, forward and backward scans for earliest and latest start times, float and the critical path, and crashing at minimum cost, with the real 2023 and 2025 VCAA questions and a short link to flow and allocation.

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How does a Leslie matrix model an age-structured population, and how do birth and survival rates project the population forward?

Set up a Leslie matrix from age-specific birth (fecundity) and survival rates, multiply it by an age-structure state matrix to project a population forward, and interpret the long-term growth and age distribution

The VCE General Mathematics Unit 4 Leslie matrix model: reading birth and survival rates from words or a life-cycle diagram, placing them in the top row and subdiagonal, projecting with S_{n+1} = LS_n, totals and rounding, working backwards, and the long-run growth rate and stable age distribution, with the real 2023 and 2025 VCAA questions.

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How do we find the shortest path between two vertices of a weighted graph, and how does Dijkstra's algorithm find the shortest path from one vertex to every other vertex?

Determine the shortest path between two specified vertices in a graph, digraph or network by inspection, and use Dijkstra's algorithm to find the shortest path between a given vertex and each of the other vertices

The VCE General Mathematics Unit 4 shortest path topic. Recognising a shortest path problem, finding it by inspection on small networks, running Dijkstra's algorithm with tentative and permanent labels, directed networks, shortest paths to every vertex, road closures and required stops, and how it differs from a minimum spanning tree.

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How do communication and dominance matrices use a matrix of zeros and ones to count direct and indirect links, and how do you rank competitors?

Build a binary communication or dominance matrix, square it to count two-step links, add the matrix and its square to combine one-step and two-step connections, and rank competitors by their total

The VCE General Mathematics Unit 4 communication and dominance matrices: reading sender and receiver, tracing a message, what an element of the square counts, one-step plus two-step totals, ranking a round-robin tournament and filling in missing results, with the real 2023 and 2025 VCAA questions.

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How are matrices added, subtracted, multiplied and raised to powers, what do the elements of a product mean in context, and when does a matrix have an inverse?

Perform elementary matrix operations (sum, difference, multiplication by a scalar, product and power), interpret them in context, and find the determinant and inverse of a matrix, including the condition for a matrix to have an inverse

The VCE General Mathematics Unit 4 matrix operations: when a sum or product is defined, the row-by-column rule, reading a product in context (totals, revenue, scaling with a diagonal matrix), powers of a square matrix, the determinant and inverse of a 2x2 matrix, the condition for an inverse, and matrix equations, with real 2023 and 2025 VCAA questions.

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What are the order, elements and types of a matrix, and how do we use matrices to store tabular information and construct a matrix from a rule for its elements?

Use the order of a matrix, element notation and the types of matrices (row, column, square, diagonal, symmetric, triangular, zero, binary, permutation and identity) and the transpose, represent tabular information as a matrix, and construct a matrix from a rule for its ij element

The VCE General Mathematics Unit 4 foundations of matrices. Order and element notation, storing a table as a matrix and interpreting an element, every matrix type named in the study design, the transpose and symmetric matrices, and building a matrix from a rule for its ij element, with real VCAA 2023 and 2025 questions.

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What are the key terms of graph theory, and how are spanning trees and shortest paths found in a weighted network?

Use graph and network terminology (vertices, edges, degree, connected, planar), apply Euler's formula and the handshake result, find a minimum spanning tree, and determine the shortest path through a weighted network

The VCE General Mathematics Unit 4 graph fundamentals: vertices, edges, degree, loops, bridges, connected, complete, bipartite and planar graphs, the sum of degrees, Euler's rule v + f = e + 2, adjacency matrices, trees and spanning trees, and a first look at minimum spanning trees and shortest paths, with the real 2023 and 2025 VCAA questions.

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How is an allocation problem represented by a bipartite graph or cost table, and how does the Hungarian algorithm find the optimum assignment of people or machines to tasks?

Use a bipartite graph and its tabular or matrix form to represent a matching problem, and determine the optimum assignment of people or machines to tasks by inspection or by the Hungarian algorithm for larger problems

The VCE General Mathematics Unit 4 matching topic. Bipartite graphs and cost tables, allocation by inspection, every step of the Hungarian algorithm (row and column reduction, covering zeros with the fewest lines, the adjustment step, reading off one zero per row and column), multiple optimal answers and maximisation, with real 2023 and 2025 VCAA questions.

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