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Question
If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A–1) = (det A)k
Solution
As we know that
`A^(-1) = (Adj A)/|A|`
`:. |A^(-1)| = |Adj A|/|A|`
= `|A|^(3-1)/|A|` [∵ If A is a non singular matrix of order n, then `|adj(A)| = |A|^(n-1)`]
= `|A|^2/|A|`
= |A|
As we are given that `|A^(-1)| = |A|^k`
∴ k = 1
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