1252 B Some Useful Mathematics
n−1byn−1 determinant. To determine the sign of a given term in the expansion,
count the number of steps of one row or one column required to get from the upper
left element to the element whose minor is desired. If the number of steps is odd, the
sign is negative. If the number of steps is even (including zero), the sign is positive.- Repeat the entire process with each determinant in the expansion until you have a
sum of 2 by 2 determinants, which can be evaluated by Eq. (B-98).
Expandinga3by3determinant gives six terms, as follows:∣ ∣ ∣ ∣ ∣ ∣
A 11 A 12 A 13
A 21 A 22 A 23
A 31 A 32 A 33
∣ ∣ ∣ ∣ ∣ ∣
A 11
∣
∣
∣
∣
A 22 A 23
A 32 A 33
∣
∣
∣
∣−A^12
∣
∣
∣
∣
A 21 A 23
A 31 A 33
∣
∣
∣
∣+A^13
∣
∣
∣
∣
A 21 A 22
A 31 A 32
∣
∣
∣
∣
A 11 (A 22 A 33 −A 23 A 32 )−A 12 (A 21 A 33 −A 23 A 31 )
+A 13 (A 21 A 32 −A 22 A 31 )
A 11 A 22 A 33 −A 11 A 23 A 32 +A 12 A 21 A 33 −A 12 A 23 A 31
+A 13 A 21 A 32 −A 13 A 22 A 31 (B-99)
Expanding larger determinants by hand can be tedious. Computer programs such as
Mathematica perform the manipulations automatically.
Determinants have a number of useful properties:Property 1. If two rows of a determinant are interchanged, the result is a determinant whose
value is the negative of the original determinant. The same is true if two columns are interchanged.Property 2. If two rows or two columns of a determinant are identical, the determinant has
value zero. This property follows from Property 1, since only zero is equal to its own negative.Property 3. If each element in one row or one column of a determinant is multiplied by the
same quantitycthe value of the new determinant isctimes the value of the original determinant.
Therefore, if annbyndeterminant has every element multiplied byc, the new determinant is
cntimes the original determinant.Property 4. If every element in any one row or in any one column of a determinant is equal
to zero, the value of the determinant is equal to zero.Property 5. If any row is replaced, element by element, by that row plus a constant times
another row, the value of the determinant is unchanged. The same is true for two columns. For
example, ∣
∣∣
∣∣
∣
∣a 11 +ca 12 a 12 a 13
a 21 +ca 22 a 22 a 23
a 31 +ca 32 a 32 a 33∣
∣∣
∣∣
∣
∣∣
∣∣
∣∣
∣
∣a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33∣
∣∣
∣∣
∣
∣(B-100)Property 6. The determinant of a triangular matrix (atriangular determinant) is equal to the
product of the diagonal elements. For example,
∣
∣∣
∣
∣∣
∣a 11 00
a 21 a 22 0
a 31 a 32 a 33∣
∣∣
∣
∣∣
∣a 11 a 22 a 33 (B-101)A diagonal determinant is a special case of a triangular determinant, so it also obeys this relation.