Matrix induction
Objective. Prove formulae for powers of matrices.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Matrix induction proves a formula for Mⁿ, the nth power of a matrix M, holds for all positive integers n, following the same base-case-and-step structure as numerical induction but using matrix multiplication. The inductive step multiplies the assumed formula for M^k by M once more, using standard matrix multiplication rules, to show the result matches the proposed formula with k replaced by k+1. For example, if M = [[1,1],[0,1]], one proves Mⁿ = [[1,n],[0,1]] by multiplying [[1,k],[0,1]] by M.
Worked example
Prove by induction that if M = [[2,0],[0,3]], then Mⁿ = [[2ⁿ,0],[0,3ⁿ]] for all positive integers n.
- 1Base case n=1: M¹ = [[2,0],[0,3]], which matches [[2¹,0],[0,3¹]]
- 2Assume M^k = [[2^k,0],[0,3^k]] for some integer k
- 3Then M^(k+1) = M^k × M = [[2^k,0],[0,3^k]] × [[2,0],[0,3]] = [[2^k×2,0],[0,3^k×3]] = [[2^(k+1),0],[0,3^(k+1)]]
- 4This matches the required form, so by induction Mⁿ = [[2ⁿ,0],[0,3ⁿ]] for all positive integers n
Answer: Proved: Mⁿ = [[2ⁿ,0],[0,3ⁿ]] by induction
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Practice
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Start practisingCommon mistake. Students multiply matrices in the wrong order during the inductive step, computing M×M^k instead of M^k×M, which can give a different result if matrices don't commute. Always multiply on the correct side matching how M^(k+1) is defined.
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