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Question
Prove that (A–1)′ = (A′)–1, where A is an invertible matrix.
Solution
Since A is an invertible matrix, so it is non-singular
We know that |A| = |A′|.
But |A| ≠ 0.
So |A′| ≠ 0 i.e. A′ is invertible matrix.
Now we know that AA–1 = A–1A = I.
Taking transpose on both sides, we get
(A–1)′A′ = A′(A–1)′
= (I)′
= I
Hence (A–1)′ is inverse of A′
i.e., (A′)–1 = (A–1)
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