Obtain a repeated-transition result without matrix software — Beyond Your Reach
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Operator-authored reference; no visitor notes are included.
Operator-authored worked artifact. Suppose a process alternates two states with transition matrix A=[[0,1],[1,0]]. You need A^1001 applied to v=[2,5], but cannot run a matrix package. Since A^2=I, A^1001=A, so the result is [5,2]. Verification: one application swaps the entries; two restore them; every odd exponent swaps. Required specification is the exact matrix, exponent and vector. This algebraic substitute avoids computing 1001 products. The same shortcut does not apply to an arbitrary matrix; first establish the identity and its assumptions.