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InterviewSolution
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If A is a 3 × 3 matrix, then what will be the value of k if Det(A-1) = (Det A)k? |
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Answer» We are given that, Order of matrix = 3 × 3 Det(A-1) = (Det A)k An n-by-n square matrix A is called invertible if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. We know that, If A and B are square matrices of same order, then Det (AB) = Det (A).Det (B) Since, A is an invertible matrix, this means that, A has an inverse called A-1. Then, if A and A-1 are inverse matrices, then Det (AA-1) = Det (A).Det (A-1) By property of inverse matrices, AA-1 = I ∴, Det (I) = Det (A).Det (A-1) Since, Det (I) = 1 \(\Rightarrow\) 1 = Det (A).Det (A-1) \(\Rightarrow\) Det (A-1) = \(\frac{1}{Det(A)}\) \(\Rightarrow\) Det (A-1) = Det (A)-1 Since, according to question, Det(A-1) = (Det A)k \(\Rightarrow\) k = -1 Thus, the value of k is -1. |
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