DETERMINANTS

The idea here is to give you a brief introduction to determinants and the method used by the Merry D eterminator (NO! Not de Terminator, Arnold) to evaluate determinants.

Our aim is not to compete with the numerous books that cover this basic material. If you are really not familiar with the subject we suggest that you consult a book for more details.

DEFINITION OF DETERMINANT

Let A be an nxn (i.e. square) matrix:

A =
a1 a2 . . . an
b1 b2 . . . bn
c1 c2 . . . cn
. . . etc . .

Then we write the determinant of A, often abbrieviated to det A, as:
| a1 a2 . . . an |
| b1 b2 . . . bn |
| c1 c2 . . . cn |
| . . . etc . . |

Note that normally the determinant is written with a solid line down each side, this is not possible for us to do at this stage.

THE DETERMINANT IS THEN DEFINED AS FOLLOWS:

  • Case n = 1,
    | a1 | = a1.
  • Case n = 2,
    | a1 a2 |
    | b1 b2 | = a1b2 - b1a2.
  • Case n = 3,
    | a1 a2 a3 |
    | b1 b2 b3 |
    | c1 c2 c3 |
    = a1 | b2 b3 | - a2 | b1 b3 | + a3 | b1 b2 |
    | c2 c3 | | c1 c3 | | c1 c2 |
    = a1(b2c3 - c2b3) - a2(b1c3 - c1b3) + a3(b1c2 - c1b2), using Case 2.

    • LETS DO ONE MORE TO SEE THE PATTERN:

  • Case n = 4,
    | a1 a2 a3 a4 |
    | b1 b2 b3 b4 |
    | c1 c2 c3 c4 |
    | d1 d2 d3 d4 |
    | b2 b3 b4 | | b1 b3 b4 | | b1 b2 b4 | | b1 b2 b3 |
    = a1 | c2 c3 c4 | - a2 | c1 c3 c4 | + a3 | c1 c2 c4 | - a4 | c1 c2 c3 |
    | d2 d3 d4 | | d1 d3 d4 | | d1 d2 d4 | | d1 d2 d3 |

    We can then use Case 3 followed by Case 2 to finally evaluate the determinant.

    • We could continue, following the pattern, to define determinants for any value of n.

    EXAMPLES

    1. Evaluate

    | 1 2 |
    | 3 4 |

    This is Case n = 2, so the determinant equals (1x4 - 3x2) = -2.

    2. Evaluate
    | 2 4 -2 |
    | 0 1 2 |
    | 3 -1 4 |

    This is case n = 3. So we get

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

    To finish it off we use Case 2 on each of the 3 remaining determinants. This gives
    2(1x4 - (-1)x2) - 4(0x4 - 3x2) - 2(0x(-1)-3x1) = 2(4 +2) - 4(0 - 6) - 2(0 - 3) = 42.

    The definitions above do not give an efficient way of evaluating determinants, especially for larger determinants. There is a much better method using our old friends the ROW OPERATIONS. The method is based on the following definition and properties of determinants.

    DEFINITION: An upper (lower) triangular matrix is a matrix with all zeros below (above) the leading diagonal (i.e. the diagonal from upper left to lower right).

    For example

    | a 0 0 0 |
    | b c 0 0 |
    | d e f 0 |
    | g h i j |

    is a lower triangular matrix. Using the definition of determinant for Case n = 4, then n = 3, then n = 2, we get that this determinant

    | c 0 0 |
    = a | e f o |
    | h i j |
    = a c | f 0 |
    | i j |
    = a c f j

    i.e. the determinant equals the product of the leading diagonal elements.

    PROPERTY 1. FOR UPPER OR LOWER TRIANGULAR MATRICES THE DETERMINANT IS THE PRODUCT OF THE LEADING DIAGONAL ENTRIES.

    EXAMPLE

    | 5 6 -1 8 74 |
    | 0 8 469 9876 32 |
    | 0 0 2 -68 -56 | = 5 x 8 x 2 x 3 x 4 = 960
    | 0 0 0 3 32 |
    | 0 0 0 0 4 |

    ROW OPERATIONS ON DETERMINANTS

    We will now make use of our Dream Team friends: Changy, Timy, Addy. So if you have not already done so or are not familiar with row operations please look at Dream Team and Row Operations.

    The effect of using row operations on determinants is as follows:

    PROPERTY 2. Interchanging any two rows multiplies the determinant by -1. E.g.

    | 1 2 3 | | 4 5 6 |
    | 4 5 6 | = (-1) | 1 2 3 |
    | 7 8 9 | | 7 8 9 |

    PROPERTY 3. Multiplying one row of a determinant by a non zero constant a, multiplies the determinant by a. E.g.

    | 3 x 1 3 x 2 3 x 3 | | 1 2 3 |
    | 4 5 6 | = (3) | 4 5 6 |
    | 7 8 9 | | 7 8 9 |

    PROPERTY 4. (the D-Addy of them all!) We can add a multiple of one row to another row without affecting the value of the determinant. E.g.

    | 1 2 3 | | 1 2 3 |
    | 4 5 6 |R2 + (-4)R1 = | 0 -3 -6 |
    | 7 8 9 | | 7 8 9 |

    THE METHOD OF THE MERRY DETERMINATOR

    A good way to evaluate determinants is to change the determinant to upper or lower triangular form using row operations with Properties 2, 3 and 4. Then use Property 1.

    EXAMPLE. Evaluate

    | 1 5 2 3 |
    D = | 2 10 8 2 |
    | 0 -3 4 2 |
    | 3 15 2 1 |

    One way of proceeding follows. Using Property 3 on Row 2 gives

    | 1 5 2 3 |
    D = (2) | 1 5 4 1 |R2 + (-1)R1
    | 0 -3 4 2 |
    | 3 15 2 1 |R4 + (-3)R1

    Now use the row operations indicated on the right, (note that R1 stands for Row 1, etc. and <=> means interchange) together with the relevant properties to get

    | 1 5 2 3 |
    D = (2) | 0 0 2 -2 |R2 <=> R3
    | 0 -3 4 2 |
    | 0 0 -4 -8 |
    | 1 5 2 3 |
    = (-2) | 0 -3 4 2 |
    | 0 0 2 -2 |
    | 0 0 -4 -8 | R4 + (2)R3
    | 1 5 2 3 |
    = (-2) | 0 -3 4 2 |
    | 0 0 2 -2 |
    | 0 0 0 -12 |

    Now using Property 1
    D = (-2) x 1 x (-3) x 2 x (-12) = -144.

    Determinants pop up in all sorts of places. One useful thing is this:
    For an n x n matrix A, its inverse, A-1, exists if and only if the determinant of A is non zero.
    You can see Matrix Inverse to find out more about matrix inverses.

    Some famous names that contributed to the development of determinants are Gottfried William Leibniz, Seki Kowa, Augustin-Louis Cauchy and Charles Dodson (alias Lewis Carroll, the author of Alice in Wonderland). Smile, like a cheshire cat.....!