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प्रश्न
Prove by Mathematical Induction that (A′)n = (An)′, where n ∈ N for any square matrix A.
उत्तर
Let P(n): (A′)n = (An)′
∴ P(1): (A′) = (A)′
⇒ A' = A'
⇒ P(1) is true.
Now, let P(k) = (A')k = (Ak)'
Where k ∈ N
And P(k + 1): (A')k+1 = (A')kA'
= (A')k'A'
= (AAk)' .....(As (AB)' = B'A')
= (Ak+1)'
Thus P(1) is true and whenever P(k) is true P(k + 1) is true,
So, P(n) is true for all n ∈ N
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