Chemistry 460
Spring 2015
Dr. Jean M. Standard
April 8, 2015
Evaluating a 3×3 Determinant – The Method of Cofactors
We have previously seen that 2×2 determinants may be evaluated by simply taking the product of the diagonal
elements minus the product of the off-diagonal elements,
a b
= a ⋅ d − b ⋅c .
c d
(1)
A 3×3 determinant is evaluated by first expanding it into a sum of 2×2 determinants using a technique called the
Method of Cofactors; the 2×2 determinants
are then evaluated as shown in Eq. (1). To demonstrate the Method of
€
Cofactors, consider a 3×3 determinant of the form:
a11 a12
a 21 a 22
a 31 a 32
a13
a 23
a 33
.
(2)
The determinant may be evaluated by expansion about any row or column. For example, consider expansion about
the first row. There will be three terms€in the expansion; each term in the expansion is a 2×2 determinant multiplied
i+ j
by (−1) a ij , where a ij is one of the elements of the first row (i.e., either a11 , a12 , or a13 ). The 2×2 determinants
are obtained by striking out the ith row and jth column and using the four elements remaining.
€
For example,
column of the 3×3 determinant are
€ when the element a ij corresponds to a11 , the first row
€ and
€ first €
struck out as shown below,
€
€
.
(3)
The remaining elements form a 2×2 determinant,
a 22
a 23
a 32
a 33
.
(4)
When the element a ij corresponds to a12 , the first row and second column of the 3×3 determinant are struck out as
shown below,
€
€
€
.
(5)
2
The remaining elements form a different 2×2 determinant,
a 21 a 23
a 31 a 33
.
(6)
Finally, when the element a ij corresponds to a13 , the first row and third column of the 3×3 determinant are struck
out as shown below,
€
€
€
.
(7)
The remaining elements form another 2×2 determinant,
a 21 a 22
a 31 a 32
.
(8)
Putting this all together, the expansion of a 3×3 determinant about the first row using the Method of Cofactors yields
the following result,
€
a11 a12
a 21 a 22
a13
a 23
a 31 a 32
a 33
1+1
= (−1)
= a11 ⋅
a11 ⋅
a 22
a 23
a 32
a 33
a 22
a 23
a 32
a 33
+
− a12 ⋅
1+2
(−1)
a 21 a 23
a12 ⋅
a 21 a 23
a 31 a 33
+
a 31 a 33
+ a13 ⋅
1+3
(−1)
a 21 a 22
a 31 a 32
a13 ⋅
a 21 a 22
a 31 a 32
(9)
.
The evaluation of the 3×3 determinant may now be completed by expanding the 2×2 determinants using Eq. (1).
€
As a specific example, consider the 3×3 determinant:
2
3 1
−1 2 4
3 0 2
.
(10)
Application of the Method of Cofactors by expansion about the first row yields
2
€
3 1
−1 2 4
3 0 2
1+1
= (−1)
= 2⋅
€
⋅2⋅
2 4
0 2
2 4
0 2
− 3⋅
+
1+2
(−1)
−1 4
3 2
⋅ 3⋅
−1 4
3 2
+ 4⋅
−1 2
3 0
+
.
1+3
(−1)
⋅4 ⋅
−1 2
3 0
(11)
3
Evaluation is completed by expanding the three 2×2 determinants according to Eq. (1),
2 3 1
−1 2 4
3 0 2
= 2⋅
2 4
0 2
− 3⋅
−1 4
3 2
+ 4⋅
−1 2
3 0
= 2( 2 ⋅ 2 − 0 ⋅ 4) − 3(−1 ⋅ 2 − 3⋅ 4) + 4 (−1 ⋅ 0 − 2 ⋅ 3)
= 2 ⋅ ( 4) − 3⋅ (−14 ) + 4 ⋅ (−6)
= 8 + 42 − 24
2 3 1
−1 2 4
3 0 2
€
= 26 .
(12)