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General Mathematics
Transition matrices
Chapter 9
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RESTS ON
Convert a percentage to a decimal
FOUNDATIONS
Percent as parts per hundred
FOUNDATIONS
Multiply matrices by pairing rows with columns and summing the products
CH 8
Classify row, column, square, symmetric and zero matrices
CH 8
Add or subtract matrices entry by corresponding entry
CH 8
Recognise that matrix multiplication is not commutative
CH 8
Name terms using the chapter's zero-indexed subscript convention
FOUNDATIONS
Write a recurrence relation as a starting value and a rule
FOUNDATIONS
Apply the closed rule for the nth term of a geometric sequence
FOUNDATIONS
Calculate the determinant of a 2 × 2 matrix
CH 8
Define the inverse as the matrix that multiplies to the identity
CH 8
Simplify a fraction
FOUNDATIONS
Signed numbers
FOUNDATIONS
State the order of a matrix as rows × columns
CH 8
Express a ratio
FOUNDATIONS
Transition matrices and diagrams
Interpreting transition matrices
State recurrences and equilibrium
Leslie matrices and population models
Decide whether a proposed model is a fixed-population, one-step transition model by checking exhaustive and exclusive states, current-state dependence, square non-negative shape and column totals of 1
Translate a labelled transition matrix into a directed diagram with column states as sources, row states as destinations and non-zero entries as labelled arrows or loops
Identify a regular transition matrix by establishing that some positive integer power has every entry strictly positive, distinguishing regularity from mere reachability
Read a transition matrix with columns as the current state
Use the property that every transition-matrix column totals 1
Construct a transition matrix from stated proportions or a diagram
Raise a square matrix to a power by repeated multiplication
Predict one step ahead from a transition matrix
Percentage of a quantity
Complete or select a two- or three-state transition model whose specified state column is unchanged by one transition, using both column totals and balanced flows
Apply a recurrence of the form S_(n+1) = T S_n + B with a signed external-change column, in the stated order at every step
Recover a preceding state by first removing the external change and then left-multiplying by the inverse: S_n = T^(-1)(S_(n+1) - B)
Write the state matrix as a column indexed by applications
Write the state-matrix recurrence relation
Apply the explicit rule for the state after n steps
Identify the steady state a state sequence settles towards
Apply the 2 × 2 inverse rule
Iterate a recurrence relation to generate terms
Classify sequence behaviour and identify a limiting value
Construct and recognise an age- or stage-structured Leslie matrix with fertility rates in the first row, survival proportions on the first subdiagonal and structural zeros elsewhere
Translate between a Leslie matrix and a directed life-cycle diagram, separating survival-to-next-stage arrows from fertility-to-first-stage arrows
Generate a population state sequence by S_(n+1) = L S_n or S_n = L^n S_0, preserving the declared stage order and time-period indexing
Calculate and interpret the total population at a stated time by summing all state-column entries, without assuming the total is conserved
Identify an approximately stable stage profile by checking that corresponding components of successive non-zero state columns share one common multiplicative factor
Classify a Leslie model as increasing, decreasing, constant, cycling or not yet settled from exact finite state probes, estimating a common long-run factor where supported
Address an element by row and column subscripts, in both directions