Solving Matrices and Determinants

by 
Tristan C

Determinant (d-tûrm-nnt) n. (according to Glencloe's Algebra Two)

                    A square array of number of expressions enclosed between two parallel vertical bars

Third order determinants are plain old determinants but in a 3 x 3 matrix.

There are two methods to solve for a determinant.

    1. Expansion by minors. Observe.

                {  a    b    c }

                {  d    e    f  }

                { g    h    i   }

The minor of an element is the determinant formed when the row and 

column containing that element are deleted.

The minor of H is       {  a    b    c  }                  { a    c }

                                  {  d    e    f   }                  { d    f  }

                                  { g    h    i   }

Once with your matrix  |a    b    c|

                                   |d    e    f |  =  a |e    f| - b |d    f| + c |d    e| = answer

                                    |g    h    i|          |h    i|       |g    i|        |g    h|

The other method is called simply "repeat the first two columns." Write the first two columns on the right.

|a    b    c|      |a    b    c| a    b

|d    e    f|  =  |d    e    f| d    e                                Voila!

|g    h    i|       |g    h     i| g    h

Then...multiply by diagonals like this...

|a    b    c|      |a    b   c| a    b

|d    e    f|  =  |d    e    f| d    e                               

|g    h    i|       |g    h     i| g    h

So you have (aei)(bfg)(cdh) then...do it backwards like this...

|a    b    c|      |a    b    c| a    b

|d    e    f|  =  |d    e    f| d    e                                

|g    h    i|       |g    h     i| g    h

Here, you have (gec)(hfa)(idb).

To find the products of the first set of diagonals and then subtract the products of the second set of diagonals.

(aei) + (bfg) + (cdh) - (gec) - (hfa) - (idb)

 

Right. So. Try one yourself. 

  In a nine game span during the 2002-2003 season, Alexander Mogilny of the Toronto Maple Leafs scored at least one point in each game. Listed are the number of points scored per game during the streak.

                  

Game 1 - 2    Game 2 - 1    Game 3 - 4    Game 4 - 1    Game 5 - 2    Game 6 - 2

Game 7 - 1    Game 8 - 4    Game 9 - 3               

Try putting these point totals in a matrix. Like this.

    { 2    1    4 }

    { 1    2    2 }

    { 1    4    3 }

Then, use either method, expansion by minors or repeat the first two columns, and evaluate.

Expansion by minors;

    { 2    1    4 }            { 2    4 }  = 2 |5    7| - 3 |6    7| + 4 |6    5| = 3

    { 1    2    2 } -------- { 1    2 }        |9    8|      |-1    8|      |-1    9|

    { 1    4    3 }

or..."repeat the first two columns"

    { 2    1    4 } { 2    1 }

    { 1    2    2 } { 1    2 }

    { 1    4    3 } { 1    4 }

 (12)+(2)+(16)-(3)-(16)-(8) = 3

 

"Whatever your difficulties in mathematics, I can assure you mine are far greater"

                                                                                                        - Albert Einstein

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