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The determinant of a matrix is denoted and is
a scalar quantity (i.e., a number). This number is involved in computation
of inverse matrices (below). For the trivial case of a 1x1 matrix,
the determinant is just the number in the matrix. For a 2x2 matrix,
the determinant is easily computed as

(19) 
For a 3x3 matrix, the determinant is again easily computed, being

(20) 
Note that each term consists of the product of three factors.
The positive terms can
be obtained by starting on the top row and multiplying by elements going
down and to the right diagonally, wrapping around when hitting the edge
of the determinant. Similarly, negative terms can be obtained by
starting
on the bottom row and multiplying by elements going up and to the right
diagonally, wrapping around when hitting an edge.
Another simple case for computing determinants is that of diagonal
(or triangular) matrices, where the determinant is just the product
of the entries on the main diagonal.
For more general matrices, there is a procedure for computing
determinants which involves cofactor matrices

(21) 
where are the elements of matrix and are the
cofactors, which are times the determinant of
a smaller
matrix found by eliminating row
and column [
]. Obviously
there is a problem if we are defining a determinant in terms of other
determinants! However, we can apply these rules iteratively until
we get to 3x3 or 2x2 matrices, for which we can take determinants using the
simple rules given above. You may wish to confirm that the cofactor method,
when applied to the case of a 3x3 matrix, yields equation
(20) when the cofactors of the smaller 2x2 matrices
are evaluated using the aid of equation (19).
Next: Inverse Matrices
Up: linear_algebra
Previous: Matrices
David Sherrill
20060815