- E14
[1V2]Topics:matrix,determinant.Difficulty:*.
Prove Jacobi’s formula:
\[ \frac{d}{d a_{i,j}} \det (A) = C_{i,j}\quad , \]where \(a_{i,j}\) is the element of \(A\) in row \(i\) and column \(j\), and \(C\) is the matrix of cofactors of \(A\), which is the transpose of the adjoint matrix \({\operatorname {adj}}(A)\). Consequently, if \(F:ℝ→ℂ^{n × n}\) is differentiable, then
\[ {\frac{d}{dt}}\det F(t)={\operatorname {tr}}\left({\operatorname {adj}}(F(t))\, {\frac{dF(t)}{dt}}\right) \]where \({\operatorname {tr}}(X)\) is the trace of \(X\).
Hint: use Laplace’s expansion for the determinant.
Solution 1
EDB — 1V2
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English
Authors:
"Mennucci , Andrea C. G."
.
Bibliography
Book index
Book index
- Jacobi
- Jacobi, formula
- cofactor matrix
- matrix, cofactor ---
- adjugate matrix
- matrix, adjugate ---
- determinant , see matrix, determinant
- matrix, determinant
- Laplace
- Laplace expansion
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