Results 1 to 10 of 10

Math Help - Determinant of a 3x3 matrix

  1. #1
    Bar0n janvdl's Avatar
    Joined
    Apr 2007
    From
    Meh
    Posts
    1,630
    Thanks
    6

    Determinant of a 3x3 matrix





    I assume that we handle this according to the pic i attched? (I only took the a_{11} part as the example)

    And could someone please tell me how we use matrices to solve linear equations?
    Attached Thumbnails Attached Thumbnails Determinant of a 3x3 matrix-matrixmatrix.jpg  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,962
    Thanks
    349
    Awards
    1
    Quote Originally Posted by janvdl View Post




    I assume that we handle this according to the pic i attched? (I only took the a_{11} part as the example)

    And could someone please tell me how we use matrices to solve linear equations?
    Google "Cramer's Rule."

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Bar0n janvdl's Avatar
    Joined
    Apr 2007
    From
    Meh
    Posts
    1,630
    Thanks
    6
    Okay i found this site: Cramer's Rule

    But you need to know how to find the determinants of a 3x3 matrix... Which I am confused about.

    If someone could only explain this part to me:
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by janvdl View Post
    Okay i found this site: Cramer's Rule

    But you need to know how to find the determinants of a 3x3 matrix... Which I am confused about.

    If someone could only explain this part to me:
    Here's how to find the determinant of a 3x3 matrix. but this method (called the cofactor expansion method) can be generalized to general nxn matrices.

    i'll use a,b,c,d... instead of a_{11}, a_{12}, \cdots because those will get annoying to type after a while. hope it doesn't mess you up

    Let's say we are given a 3x3 matrix \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right)

    you begin by imagining a checker board pattern of pluses and minuses, with a + sign starting at the top left corner, that is, think of

    \left( \begin{array}{ccc} + &  - &  + \\  - &  + &  - \\ + & - &  + \end{array} \right)

    now that accounts for the signs in front of the terms that you have. this is why you had + a_{11} ... - a_{12} ... + a_{13} in your example.

    now, in using the cofactor expansion method, you can expand along ANY row OR column you wish. your example showed an expansion along the first row, but it may be convenient to expand along the second column, for instance. for example, that's what i'd do if i noticed that e = h = 0, and all other terms were non-zero, because then, expanding along the second column would require doing the determinant of only 1 2x2 matrix.

    so the determinant in such a case would be: \mbox{det} \left( \begin{array}{ccc} a &  b & c  \\ d  & e  & f  \\ g & h  & i  \end{array} \right) = -b \left| \begin{array}{cc} d & f  \\ g  & i \end{array} \right| ..........when we write a matrix with straight braces, it means the determinant of that matrix

    now we will see how they got such a formula.

    when you choose a column or row to expand along, begin with the first element in that row or column, and affix the sign in front of it as directed by the checker board pattern. upon doing so, you should eliminate every term in that element's row and column. (so if we are considering the element b, we would affix a minus sign in front of it and eliminate the elements a,c,e and h, since they fall in the same row and column of b). this will result in a new matrix that is one dimension lower, in this case, a 2x2 matrix (the matrix with row 1: d,f and row 2:g,i). find the determinant of that matrix and multiply by the original term. repeat this process for each term in that row or column you decided to expand along.

    for example, using my matrix, let's call the matrix A. finding the determinant by expanding along the third row would be:

    \mbox{det}A = g \left| \begin{array}{cc} b & c \\ e  & f \end{array} \right| - h \left| \begin{array}{cc} a & c \\ d  & f \end{array} \right| + i \left| \begin{array}{cc} a & b \\ d  & e \end{array} \right|

    expanding along the second column would be:

    \mbox{det}A = -b \left| \begin{array}{cc} d & f \\ g & i \end{array} \right| + e \left| \begin{array}{cc} a & c \\ g & i \end{array} \right| - h \left| \begin{array}{cc} a & c \\ d & f \end{array} \right|


    Here's an exercise: what would the formula be for expanding along the second row?


    did you have any problems with Cramer's rule itself?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Bar0n janvdl's Avatar
    Joined
    Apr 2007
    From
    Meh
    Posts
    1,630
    Thanks
    6
    Quote Originally Posted by Jhevon View Post
    Here's an exercise: what would the formula be for expanding along the second row?


    did you have any problems with Cramer's rule itself?
    Thanks a lot Jhevon.

    For expanding the second row, :

     \mbox{det}A = -d \left| \begin{array}{cc} b & c \\ h & i \end{array} \right| + e \left| \begin{array}{cc} a & c \\ g & i \end{array} \right| - f \left| \begin{array}{cc} a & b \\ g & h \end{array} \right|

    And no, Cramer's rule seems easy enough
    Follow Math Help Forum on Facebook and Google+

  6. #6
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by janvdl View Post
    Thanks a lot Jhevon.

    For expanding the second row, :

     \mbox{det}A = -d \left| \begin{array}{cc} b & c \\ h & i \end{array} \right| + e \left| \begin{array}{cc} a & c \\ g & i \end{array} \right| - f \left| \begin{array}{cc} a & b \\ g & h \end{array} \right|

    And no, Cramer's rule seems easy enough
    well done. that is correct

    for some reason, i don't like Cramer's rule. i don't use it unless they ask for it
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Bar0n janvdl's Avatar
    Joined
    Apr 2007
    From
    Meh
    Posts
    1,630
    Thanks
    6
    Quote Originally Posted by Jhevon View Post
    well done. that is correct

    for some reason, i don't like Cramer's rule. i don't use it unless they ask for it
    It seems easy and effective enough...
    Follow Math Help Forum on Facebook and Google+

  8. #8
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by janvdl View Post
    It seems easy and effective enough...
    i find matrix multiplication and finding inverses easier...well, i guess finding the inverse of a 3x3 might be a little tedious
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    9,962
    Thanks
    349
    Awards
    1
    Quote Originally Posted by Jhevon View Post
    i find matrix multiplication and finding inverses easier...well, i guess finding the inverse of a 3x3 might be a little tedious
    The reason I suggested it is that I'm not comfortable with inverting 3x3 (or larger) matrices. I admit it's probably simply that I haven't done it enough. Generally in Quantum Mechanics (where I initially learned how to do many of the tricks I know with matrices) you don't have to invert them, so I have never worked with it much.

    So I like Cramers because I don't have to rely on my calculator to finish the problem.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  10. #10
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by topsquark View Post
    The reason I suggested it is that I'm not comfortable with inverting 3x3 (or larger) matrices. I admit it's probably simply that I haven't done it enough. Generally in Quantum Mechanics (where I initially learned how to do many of the tricks I know with matrices) you don't have to invert them, so I have never worked with it much.

    So I like Cramers because I don't have to rely on my calculator to finish the problem.

    -Dan
    that's fine. to each his own

    This is my 55th post!!!
    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 of a matrix
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: November 10th 2010, 09:07 PM
  3. Determinant of matrix
    Posted in the Calculus Forum
    Replies: 3
    Last Post: January 25th 2010, 06:35 PM
  4. Determinant of a 3X3 matrix
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: November 17th 2009, 05:01 AM
  5. Determinant of a 2 x 3 matrix?
    Posted in the Algebra Forum
    Replies: 3
    Last Post: June 6th 2008, 01:19 PM

Search Tags


/mathhelpforum @mathhelpforum