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

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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