Can a determinant be differentiated? |
No problem! There
are two ways. They are in fact equivalent. Here we use a 3 x 3 determinant
for demonstration. You can easily extend the method to higher order
determinant. Of course, this problem is meaningful if the entries of the
determinant are not all constants. |
Method 1 You
can differentiate the first row (or column) and keep the entries of the other
row untouched. Then you can get a determinant. Do this for all other rows (or
columns). Then find the sum of all such determinants. =
5 + 4x – 12x^{2} – 6x^{5} (the
expansion of the determinants is omitted) |
Method 2 (1) Form
the matrix and differentiate all entries of the matrix: (2) Find
the cofactor matrix of A (Any
element of the minor matrix is the determinant formed by
deleting the row and the column that contain the element. The cofactor
matrix is formed by multiplying elements of minor matrix by (-1)^{i+j},
for i^{th} row and j^{th} column) : Note that
in fact some of the elements of C need not be found because the corresponding
elements in B are 0 and are marked by *. (3) The derivative
of the determinant formed by the matrix A is found by multiplying
corresponding elements of B and C and then found the sum. |A|
= 2x(-x^{4} – 4x + 2) + 1(2) + 3x^{2}
(-x^{3}) + 1(-x^{5} + 3) =
5 + 4x – 12x^{2}
– 6x^{5} |