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**Unformatted text preview: **erminant of a 3 × 3 matrix a1,1 a1,2 a1,3
A = a2,1 a2,2 a2,3 a3,1 a3,2 a3,3
is deﬁned by det(A) = a1,1 a1,2 a1,3
a2,1 a2,2 a2,3
a3,1 a3,2 a3,3 = a1,1 C1,1 − a1,2 C1,2 + a1,3 C1,3 , (10.4) where C1,1 , −C1,2 , and C1,3 are the so-called cofactors of a1,1 , a1,2 , and a1,3 ,
respectively, and are deﬁned by
C1,1 = a2,2 a2,3
,
a3,2 a3,3 C1,2 = a2,1 a2,3
,
a3,1 a3,3 and C1,3 = a2,1 a2,2
.
a3,1 a3,2 We observe that C1,j is the determinant of the 2 ×2 submatrix of A that is obtained
by deleting the 1st row and j th column of A.
Example 10.29 (determinant of a 3 × 3
Find the determinant of the matrix 10
0 1
A=
20 matrix) 1
0 .
0 10. Matrices 275 Solution: We apply formula (10.4) to compute the determinant:
101
010
200 det(A) = =1× 01
00
10
+1×
−0×
20
20
00 = 1 × (1 × 0 − 0 × 0) − 0 + 1 × (0 × 0 − 1 × 2)
= 1 × 0 + 1 × (−2) = −2. 2 Example 10.30 (determinant of a 3 × 3 matrix)
Determine the determinant of the 3 × 3 matrix 123
A = 4 6 5 .
987
Solution: We have from (10.4) that det(A) = 123
465
987 = 1× 65
45
46
−2×
+3×
87
97
98 = (6 × 7 − 5 × 8) − 2 × (4 × 7 − 5 × 9) + 3 × (4 × 8 − 6 × 9)
= (42 − 40) − 2 × (28 − 45) + 3 × (32 − 54)
= 2 − 2 × (−17) + 3 × (−22) = 2 + 34 − 66 = −30. 2 Remark 10.31 (use 3 × 3 matrix to determine the vector product)
In Remark 9.44 we mentioned the following formula for the vector product
a × b = (a1 , a2 , a3 ) × (b1 , b2 , b3 )
= ijk
a1 a2 a3
b1 b2 b3 = a2 a3
a1 a3
a1 a2
i−
j+
k
b2 b3
b1 b3
b1 b2 = (a2 b3 − a3 b2 ) i − (a1 b3 − a3 b1 ) j + (a1 b2 − a2 b1 ) k,
which we can now understand with our knowledge of the determinant notation.
Now we learn the formula for the determinant of an n × n matrix. This formula is
a generalization of the formula (10.4) for the determinant of a 3 × 3 matrix. 276 10. Matrices Deﬁnition 10.32 (determinant of an n × n matrix)
Let A = (ai,j ) be an n × n matrix whose entry in ith row and j th column is denoted
by ai,j . Then the determinant of the n × n matrix A is given by
n ai,j (−1)i+j Ci,j , det(A) = (10.5) j =1 for any i ∈ {1, 2, . . . , n}, where Ci,j is the determinant of the (n − 1) × (n − 1) matrix obtained by deleting the ith row and j th column from A. The term (−1)i+j Ci,j
is called the cofactor of ai,j .
Remark 10.33 (comments on the determinant)
(a) The notion ‘for any i ∈ {1, 2, . . . , n}’ in Deﬁnition 10.32 implies that for any
choice of i the value of the determinant det(A) is the same.
(b) In the above deﬁnition, we have ‘expanded about the ith row’. We can also
‘expand about the j th column’. Indeed, the determinant of A is also given by
n ai,j (−1)i+j Ci,j , det(A) = (10.6) i=1 for any j ∈ {1, 2, . . . , n}.
(c) For a 3 × 3 matrix a1,1 a1,2 a1,3
A = a2,1 a2,2 a2,3 ,
a3,1 a3,2 a3,3 expanding about the ﬁrst row (that is, taking i = 1 in Deﬁnition 10.32) gives
that the determinant of A is
3 a1,j (−1)1+j C1,j det(A) =
j =1 = a1,1 (−1)1+1 C1,1 + a1,2 (−1)1+2 C1,2 + a1,3 (−1)1+3 C1,3
= a1,1 C1,1 − a1,2 C1,2 + a1,3 C1,3 ,
where
C1,1 = a22 a23
,
a32 a33 C1,2 = a21 a23
,
a31 a33 and C1,3 = a21 a22
.
a31 a32 This is the formula (10.4) given in Deﬁnition 10.28, and we see that Deﬁnition
10.28 is just a special case of Deﬁnition 10.32. 10. Matrices 277 We now come b...

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