# If A and B are 4x4 matrices, det(A) = -4, det(B) = 3, then (f) det(A(B^T)^-1) = ???,

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First, note that solving for determinants is distributive:

`det(AB) = det(A) det(B)`

Hence,

`det(A(B^T)``^-1) = det(A) det((B^T)^-1)`

` `

Then, note the following:

i. The determinant of the inverse of a matrix, is the reciprocal of the determinant of the original matrix;

ii. The determinant of the transpose of a matrix is just the determinant of the original matrix.

Hence,

`det((B^T)^-1) = 1/det(B^T) = 1/det(B) = 1/3`

Therefore, the answer you are looking for is:

`det(A(B^T)^-1) = -4 * (1/3) = -4/3.`

We know

`det(XY)=det(X)det(Y)`

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

`therefore`

`det(A(B^T)^(-1))=det(A)det((B^T)^(-1))`

But

`det(B^T)=det(B)`

`det((B^T)^(-1))=det(B^(-1))`

`therefore`

`det(A(B^T)^(-1))=det(A)det(B^(-1))`

`=det(A)/det(B)`

`=(-4)/3`

`` Ans.