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Suppose that the eigenvalues of matrix $A$ are $1, 2, 4$. The determinant of $\left(A^{-1}\right)^{T}$ is _________.
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Determinant of Matrix A = product of eigen values = $1 \times 2 \times 4 =8$

Determinant of Inverse Matrix of A, $\text{det}(A^{-1}) = \frac{1}{\text{det}(A)} =\frac{1}{8}$

Determinant remains same after the Transpose

so, Determinant of $(A^{-1})^T$ = $\text{det}(A^{-1}) = \frac{1}{8}$ = 0.125

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0.125

Eigen value of A inverse is 1,1/2,1/4. Product of those eigen values gives determinant value.

Transposing a matrix doesn't change is eigen value.