Here (1 ,2) is an even permutation of (1 ,2) and (2 ,1) represents an odd permuta-
tion of (1 ,2) A mnemonic device to remember this 2 × 2 determinant is illustrated by
the following figure
Example 10-16. The 3 × 3 matrix A=
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33has the determinant
|A|=∣∣
∣∣
∣∣a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣∣
∣∣
∣∣= + a^11 a^22 a^33 +a^12 a^23 a^31 +a^13 a^21 a^32
−a 13 a 22 a 31 −a 11 a 23 a 32 −a 12 a 21 a 33A mnemonic device to remember this 3 × 3 determinant is to append the first two
columns to the end of the matrix and draw diagonal lines through the elements to
create the following figure, where the elements on each diagonal are multiplied.
Note that the determinant of a n×nmatrix has n-factorial terms and consequently
if nis large, then mnemonic devices like those above are not employed because the
calculations become cumbersome and sometimes extremely lengthy. Instead it has
been found that by using row reduction methods^2 the given matrix can be converted
to an equivalent upper triangular or lower triangular matrix having all zeros either
below or above the main diagonal. The determinant of these special triangular
matrices is then just a product of the diagonal elements.
Example 10-17. Find the derivative of the determinant
y=det A=|A|=
∣∣
∣∣a^11 (t) a^12 (t)
a 21 (t) a 22 (t)∣∣
∣∣(^2) Row reduction methods are considered later in this chapter.