Results 1 to 3 of 3

Math Help - Derivative of a determinant

  1. #1
    Junior Member
    Joined
    Aug 2009
    Posts
    30

    Derivative of a determinant

    Which is a formula for calculating the derivative of a determinant with respect to one of its elements? How do I prove this?

    I know the result has something to do with the inverse, but I can't just quite grasp the proof and can't find it anywhere
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor red_dog's Avatar
    Joined
    Jun 2007
    From
    Medgidia, Romania
    Posts
    1,252
    Thanks
    5
    Let D(x)=\begin{vmatrix}f_{11}(x) & f_{22}(x) & \dots & f_{1n}(x)\\f_{21}(x) & f_{22}(x) & \ldots & f_{2n}(x)\\\ldots & \ldots & \ldots & \ldots\\f_{i1}(x) & f_{i2}(x) & \ldots & f_{in}(x)\\\ldots & \ldots & \ldots & \ldots\\f_{n1}(x) & f_{n2}(x) & \ldots & f_{nn}(x)\end{vmatrix}

    Then

    D'(x)=\sum_{i=1}^n\begin{vmatrix}f_{11}(x) & f_{22}(x) & \dots & f_{1n}(x)\\f_{21}(x) & f_{22}(x) & \ldots & f_{2n}(x)\\\ldots & \ldots & \ldots & \ldots\\f_{i1}'(x) & f_{i2}'(x) & \ldots & f_{in}'(x)\\\ldots & \ldots & \ldots & \ldots\\f_{n1}(x) & f_{n2}(x) & \ldots & f_{nn}(x)\end{vmatrix}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2009
    Posts
    91
    If \mathbf A is an invertible matrix which depends on a real parameter t, and if \frac{\mathrm d\mathbf A}{\mathrm dt} exists, then

    \frac{\mathrm d}{\mathrm dt}(\det\mathbf A)=(\det\mathbf A)\mathop{\textrm{tr}}\biggl(\frac{\mathrm d\mathbf A}{\mathrm dt}\mathbf A^{-1}\biggr), where \mathop{\textrm{tr}}(\mathbf B) is the trace of \mathbf B.

    I have discovered a truly marvellous proof of the above statement, which unfortunately my brain is too small to contain.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Derivative of a matrix inverse and matrix determinant
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 24th 2011, 08:18 AM
  2. Determinant help
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: January 16th 2010, 12:55 AM
  3. Replies: 1
    Last Post: January 1st 2010, 10:37 AM
  4. Determinant 3
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 28th 2009, 11:53 AM
  5. Determinant
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: January 21st 2009, 06:27 PM

Search Tags


/mathhelpforum @mathhelpforum