# Let A be an invertible 3 by 3 matrix. Assume that (P)(A^2)(P^-1) = 10(A^T)(A^2) Then det(A) = ??

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`det(PA^2P^(-1))=det(P)det(A^2)det(P^(-1))`

`` `det(P^(-1))=1/(det(P))`

Thus

`det(PA^2P^(-1))=det(A^2)`

Thus

`det(PA^2P^(-1))=det(10A^TA^2)=det(A^2)`

`det(10A^T)det(A^2)=det(A^2)`

since A is invertible therefore det(A) is non vanishing.i.e

`det(A)!=0`

`det(10A^T)=1`

`10^3det(A^T)=1` (order of A is 3x3)

But `det(A^T)=det(A)`

`therefore det(A)=1/10^3`

`det(A)=10^(-3)`

Ans.