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 – 12x2 – 6x5           (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 ith row and jth 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(-x4 – 4x + 2) + 1(2) + 3x2 (-x3) + 1(-x5 + 3)

=     5 + 4x – 12x2 – 6x5
The text passage discusses the differentiation of determinants and provides two methods for doing so, including an example calculation.
no_question

the passage does not contain a mathematical question but rather provides an explanation and methods for differentiating determinants. the example calculation provided is part of the explanation and not a separate question.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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 – 12x2 – 6x5           (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 ith row and jth 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(-x4 – 4x + 2) + 1(2) + 3x2 (-x3) + 1(-x5 + 3)

=     5 + 4x – 12x2 – 6x5
user
The text passage discusses the differentiation of determinants and provides two methods for doing so, including an example calculation.

TYPE: NO_QUESTION

The passage does not contain a mathematical question but rather provides an explanation and methods for differentiating determinants. The example calculation provided is part of the explanation and not a separate question.