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 I_{n} 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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