On Differentiating a Matrix

The theorem is well known. So also is the theorem that if concerning a determinant Λ and its reciprocal expressed by means of cofactors A ij of a ij . Not quite so well known is the Cayley Hamilton theorem that a matrix X =x ij satisfies its own characteristic equation Unlike as these three results are, they nevertheless can be looked upon as particular phases of a general theorem concerning a matrix differential operator acting upon a function of a matrix X or its transposed.