I need to prove by induction that this 2x2 matrix: [cosθ -senθ; senθ cosθ]^n is equal to this 2x2 matrix: [cos(n)θ -sen(n)θ; sen(n)θ cos(n)θ] for all n ∈ Z. Since it's by induction, I've already got the and , but... now what? Thanks!
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Originally Posted by El Joey I need to prove by induction that this 2x2 matrix: [cosθ -senθ; senθ cosθ]^n is equal to this 2x2 matrix: [cos(n)θ -sen(n)θ; sen(n)θ cos(n)θ] for all n ∈ Z. Since it's by induction, I've already got the and , but... now what? Thanks! Here's the idea: Let Show that , by multiplying out. Now choose and use the induction hypothesis to finish the proof
Hello, Originally Posted by El Joey I need to prove by induction that this 2x2 matrix: [cosθ -senθ; senθ cosθ]^n is equal to this 2x2 matrix: [cos(n)θ -sen(n)θ; sen(n)θ cos(n)θ] for all n ∈ Z. Since it's by induction, I've already got the and , but... now what? Thanks! Let Assume that Then Now make the multiplication and use these formula : and you're done
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