Reading a transition matrix
A transition matrix records the proportion moving from each current state to each next state. Read a column for one source state, multiply by its current count and add all contributions that arrive at the required state
\[S_1=TS_0\]
- Read source columns For an entry in row i and column j, column j is the current state and row i is the next state \(\displaystyle t_{21}=0.25\Rightarrow\text{25\% move from state 1 to state 2}\)
- Convert percentages Write a percentage as a decimal proportion before multiplying \(\displaystyle 25\%=0.25\)
- Find one contribution Multiply a transition proportion by the current count in its source state \(\displaystyle 0.25\times80=20\)
- Add all arrivals Add the contributions from every current state that can reach the destination state \(\displaystyle 0.25\times80+0.80\times50=60\)
Worked example
A two-state population has 72 people in P and 45 people in Q. The transition matrix has columns for current states and rows for next states. Find the number in Q after one step
\[T=\begin{bmatrix}0.75&0.20\\0.25&0.80\end{bmatrix},\quad S_0=\begin{bmatrix}72\\45\end{bmatrix}\]
- \(\displaystyle t_{21}=0.25\)The entry in row 2, column 1 gives the proportion moving from P to Q.
- \(\displaystyle 0.25\times72=18\)The contribution from P to Q is 18 people
- \(\displaystyle 0.80\times45=36\)The contribution staying in Q is 36 people
- \(\displaystyle S_1(Q)=18+36=54\)Adding both arrivals gives 54 people in Q after one step
Common mistakes
Reversing row and column meanings
✕\(\displaystyle t_{12}=0.20\quad(P\to Q)\)✓\(\displaystyle t_{21}=0.25\quad(P\to Q)\)
Rows identify destination states and columns identify source states, so reversing them assigns each movement to the wrong transition entryCounting only stayers
✕\(\displaystyle S_1(Q)=0.80\times50=40\)✓\(\displaystyle S_1(Q)=0.25\times80+0.80\times50=60\)
The next count in a state includes every contribution arriving there, including movement from other statesWriting a percentage as a whole number
✕\(\displaystyle 25\times80=2000\)✓\(\displaystyle 0.25\times80=20\)
A percentage is divided by 100 before it is used as a transition proportionDifficulty levels
introRead a percentage transition and calculate one movement between two states
standardCombine a movement contribution and a stay contribution to find a next-state count
challengeCalculate all next-state totals in a three-state model and check conservation of the total
examChoose the transition interpretation or calculation that uses the stated matrix convention
Questions about this topic
Which direction does an entry in row i and column j represent?+
Column j is the current source state and row i is the next destination state, so the entry gives the proportion moving from j to i
Why are several products added for one next state?+
People in the next state can come from several current states, so each source contribution is calculated and then added
Why must the transition percentages in a column total 100%?+
A column describes all possible next states for one current state, so their proportions must account for the whole current group
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 4: Matrices
In this chapter
State the order of a matrix and locate its elements4A Matrix types and order: Identify row, column, square, symmetric and zero matricesRead information from a labelled data matrix4B Matrix addition and subtraction: Identify row, column, square, symmetric and zero matricesAdd and subtract matrices4B Matrix addition and subtraction: Find unknown entries in a matrix equationMultiply a matrix by a scalarCombine scalar multiples of matricesDecide whether a matrix product is defined and give its orderMultiply matrices4D Matrix products and powers: Evaluate powers of a square matrixIdentify and construct the identity matrix4E Inverse matrices and simultaneous equations: Find unknown entries in a matrix equationFind the determinant of a 2 × 2 matrixFind the inverse of a 2 × 2 matrixWrite simultaneous equations as a matrix equationSolve simultaneous equations using the inverse matrix4F Matrices for networks: Evaluate powers of a square matrixRepresent a network in a matrixIdentify the network a connection matrix describesInterpret connection sums and symmetryCount routes using powers of a connection matrixSet up a transition matrixUse a transition matrix to predict the next step4H State matrices and recurrence: Evaluate powers of a square matrixGenerate state matrices by recursionDetermine the long-run state of a transition modelUse matrix multiplication to answer an application questionUse row and column matrices of ones to extract totals