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**Unformatted text preview: **ack to the matrix from Example 10.30, and we compute its determinant in two diﬀerent ways.
Example 10.34 (determinant of 3 × 3 matrix)
Consider again the matrix A from Example 10.30, given by 123
A = 4 6 5 .
987
(a) Expanding about the second row (that is, taking (10.5) with i = 2) gives that
the determinant of A is
3 a2,j (−1)2+j C2,j det(A) =
j =1 = a2,1 (−1)2+1 C2,1 + a2,2 (−1)2+2 C2,2 + a2,3 (−1)2+3 C2,3
= −a2,1 C2,1 + a2,2 C2 2 − a2,3 C2,3
= −4 C2,1 + 6 C2,2 − 5 C2,3 ,
where
C2,1 = 23
87 = 14 − 24 = −10, C2,2 = 13
97 = 7 − 27 = −20, C2,3 = 12
98 = 8 − 18 = −10. Hence
det(A) = −4 C2,1 + 6 C2,2 − 5 C2,3
= −4 × (−10) + 6 × (−20) − 5 × (−10)
= 40 − 120 + 50 = −30,
agreeing with the answer we obtained previously in Example 10.30.
(b) Expanding about the ﬁrst column (that is, taking (10.6) with j = 1) gives that
the determinant of A is
3 ai,1 (−1)i+1 Ci,1 det(A) =
i=1 278 10. Matrices
= a1,1 (−1)1+1 C1,1 + a2,1 (−1)2+1 C2,1 + a3,1 (−1)3+1 C3,1
= a1,1 C11 − a2,1 C2,1 + a3,1 C3,1
= C1,1 − 4 C2,1 + 9 C3,1, where
C1,1 = 65
87 = 42 − 40 = 2, C2,1 = 23
87 = 14 − 24 = −10, C3,1 = 23
65 = 10 − 18 = −8. Hence
det(A) = C1,1 − 4 C2,1 + 9 C3,1 = 2 − 4 × (−10) + 9 × (−8) = 2 + 40 − 72 = −30,
again agreeing with the answer we obtained previously in Example 10.30. 2 In the next lemma we get information about how the determinant changes if we
perform certain operations on the matrix, such as interchanging rows (or columns)
of the matrix, multiplying a row (or column) with a scalar, or even adding a multiple
of one row (or column) to another row (or column). Such operations can be used
to modify the the matrix (whose determinant we want to compute) and thus may
allow a much easier computation of the determinant of the modiﬁed matrix.
Lemma 10.35 (properties of the determinant)
We can use the following properties of the determinant when ﬁnding the
determinant of an n × n matrix:
(i) multiplying all the elements of a single row or column of a matrix by a real number λ results in the determinant of the matrix being
multiplied by λ;
(ii) interchanging any two rows or any two columns of a matrix changes
the sign of the determinant of the matrix;
(iii) adding a multiple of one row (or column) of a matrix to another
row (or column) does not change the determinant of the matrix;
(iv) if any row or column of a matrix consists entirely of zeros, then the
determinant of the matrix is zero;
(v) if any two rows or any two columns of a matrix are the same, then
the determinant of the matrix is zero. 10. Matrices 279 Example 10.36 (determinant of a 4 × 4
Find the determinant of the matrix 4
3 7
9
A=
0
4
3 −1 matrix)
0
2
2
4 1
3
1
0 . Solution: Rule (i) allows us to take out the common factor 2 from column 3:
4
3
9
7
det(A) = 2 ×
4
0
3 −1 0
1
1
2 Next, we use rule (ii) to interchange columns 1 and
left corner 1:
1
3
3
7
det(A) = 2 × (−1) ×
1
0
0 −1 1
3
.
1
0
4 to make the entry in the top
0
1
1
2 4
9
.
4
3 We now apply rule (iii) twice. Firstly, we subtract three times row 1 from row 2
new
(that is, r2 = r2 − 3 r1 ):
1
3
0 −2
det(A) = −2 ×
1
0
0 −1 0
4
1 −3
.
1
4
2
3 new
Now, we subt...

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