ITPEC FE Subject B October 2024 Question 15

Matrix Exponentiation (Markov Chain) — multiply a transition matrix to find state probabilities after multiple steps.

The program computes P³ (transition matrix raised to the 3rd power) to predict weather probabilities 3 days ahead.

Blank A — loop count:
- dmat starts as pmat (= P¹)
- Each iteration multiplies dmat × pmat, so after n iterations: dmat = P^(n+1)
- To get P³, we need 2 iterations → m from 1 to 2 → A = 2

Blank B — matrix multiplication formula:
- Standard: C[i,j] = Σ(k) A[i,k] × B[k,j]
- Here: smat[i,j] += dmat[i, k] × pmat[k, j]

Answer: (a) — A = 2, B = dmat[i, k] × pmat[k, j]

Why not others:
- (b) — wrong index order (dmat[k,i] × pmat[j,k]), not valid matrix multiplication
- (c) — A=3 would give P⁴, not P³
- (d) — A=3 with wrong index order
- (e) — A=4 would give P⁵
- (f) — A=4 with wrong index order

Key rule: for matrix multiplication C = A × B, always use C[i,j] += A[i,k] × B[k,j] — the shared index k is the inner dimension.

AI-generated — may contain errors