Q. 564.3( 6 Votes )

If A is a 3 × 3 matrix, then what will be the value of k if Det(A-1) = (Det A)k?

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


1 = Det (A).Det (A-1)



Det (A-1) = Det (A)-1


Since, according to question,


Det(A-1) = (Det A)k


k = -1


Thus, the value of k is -1.


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