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Determinants: 33 Determinants (page 2 of 2)

Sections: 22 determinants, 33 determinants


The computations for 33 determinants are messier than for 22's. Various methods can be used, but the simplest is probably the following:   Copyright Elizabeth Stapel 2004-2011 All Rights Reserved

     
    Take a matrix
    A:

     

    [[ 1 2 3 ][ 0 -4 1 ][ 0 3 -1]]

      

     
    Write down its determinant:

     

    || 1 2 3 || 0 -4 1 || 0 3 -1 ||

      

     
    Extend the determinant's grid by rewriting the first two columns of numbers:

     

    ( 1 2 3 1 2 )( 0 -4 1 0 -4 )( 0 3 -1 0 3 )

       

     
    Then multiply along the down-diagonals:

     

    multiplying down

      

     

    ...and along the up-diagonals

     

    multiplying up

      

     

    Add the down-diagonals and subtract the up-diagonals:

     

    det(A) = (4) + (0) + (0) - (0) - (3) - (0)

      

     
    And simplify:

     

    det(A) = (4) + (0) + (0) - (0) - (3) - (0) = 4 - 3 = 1

      Then det(A) = 1.

  • Find the deteriminant of the following matrix:
    • [[ 5 –2 1 ][ 0 3 –1 ][ 2 0 7 ]]
        

    First I convert from the matrix to its determinant, with the extra columns:

     

    ( 5 –2 1 5 –2 )( 0 3 –1 0 3 )( 2 0 7 2 0 )

       

     

    Then I multiply down and up the diagonals:

     

    multiplications along the diagonals
      

    Then I add the down-diagonals, subtract the up-diagonals, and simplify for the final answer:

      || 5 –2 1 || 0 3 –1 || 2 0 7 || = (105) + (4) + (0) – (6) – (0) – (0) = 109 – 6 = 103

There are other methods for simplifying determinants by hand, and these other methods are required when evaluating larger determinants by hand, but those methods can probably wait until later. For the time being, note that your graphing calculator should be able to evaluate the determinant of any (square) matrix you enter. For instance:

    [C] = [[ 5 –2 1 ][ 0 3 –1 ][ 2 0 7 ]] then det([C]) = 103

But make sure, even if you have a graphing calculator, that you can evaluate 22 and 33 determinants, because you are likely to have word problems where the determinants contain variables that your calculator can't handle.

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Cite this article as:

Stapel, Elizabeth. "Determinants: 3x3 Determinants." Purplemath. Available from
    http://www.purplemath.com/modules/determs2.htm. Accessed
 

 



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