Gradients of the Determinant

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Linear Algebra › Gradients of the Determinant

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Which of the following expressions is one for the gradient of the determinant of an $n \times n$ matrix ?

$(adj(A))^{-1}$

None of the other answers

$\mathbf{x}^{-1}A\mathbf{x}, (|\mathbf{x}|=1)$

Explanation

The expression for the determinant of using co-factor expansion (along any row) is

$|A| = \sum_{i=1}^n(-1)^{i+j}A_{ij}M_{ij}$

In order to find the gradient of the determinant, we take the partial derivative of the determinant expression with respect to some entry $A_{kl}$ in our matrix, yielding $(-1)^{k+l}M_{kl} = (adj(A))_{kl} = (adj(A))^T$ .

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