414 CHAPTER 19. EIGENVALUES AND EIGENVECTORS
Example 19.2.19 Let A =
(1 23 4
)and let A1 =
(3 41 2
). detA =−2, det(A1) = 2.
Theorem 19.2.20 Let A be an n× n matrix and let A1 be a matrix which resultsfrom applying row operation 3. That is you replace some row by a multiple of another rowadded to itself. Then det(A) = det(A1).
Example 19.2.21 Let A =
(1 23 4
)and let A1 =
(1 24 6
). Thus the second row of A1
is one times the first row added to the second row. det(A) =−2 and det(A1) =−2.
Theorem 19.2.22 In Theorems 19.2.16 - 19.2.20 you can replace the word, “row”with the word “column”.
There are two other major properties of determinants which do not involve row opera-tions overtly.
Theorem 19.2.23 Let A and B be two n×n matrices. Then
det(AB) = det(A)det(B).
Also,det(A) = det
(AT).
Example 19.2.24 Compare det(AB) and det(A)det(B) for
A =
(1 2−3 2
),B =
(3 24 1
).
First
AB =
(1 2−3 2
)(3 24 1
)=
(11 4−1 −4
)and so
det(AB) = det(
11 4−1 −4
)=−40.
Now
det(A) = det(
1 2−3 2
)= 8, det(B) = det
(3 24 1
)=−5.
Thus det(A)det(B) = 8× (−5) =−40.
19.2.4 Finding Determinants Using Row OperationsTheorems 19.2.20 - 19.2.22 can be used to find determinants using row operations. Aspointed out above, the method of Laplace expansion will not be practical for any matrix oflarge size. Here is an example in which all the row operations are used.
Example 19.2.25 Find the determinant of the matrix
A =
1 2 3 45 1 2 34 5 4 32 2 −4 5