خواص المحددات

This lecture we will start studying a properties of determinants, and algorithms of computing
them. Let’s recall, that we defined a deTheorem 1.1 (1st elementary row operation). If 2 rows of a matrix A are interchanged,
then the determinant changes its sign.
Proof. Suppose B arises from A by interchanging rows r and s of A, and suppose r < s. Then
we have that brj = asj and bsj = arj for any j, and aij = bij if i 6= r; s. Now
detB =
X
all permutations of n elements ¾
sgn(¾)b1¾(1) ¢ ¢ ¢ br¾(r) : : : bs¾(s) : : : bn¾(n)
=
X
all permutations of n elements ¾
sgn(¾)a1¾(1) ¢ ¢ ¢ as¾(r) : : : ar¾(s) : : : an¾(n)
=
X
all permutations of n elements ¾
sgn(¾)a1¾(1) ¢ ¢ ¢ ar¾(s) : : : as¾(r) : : : an¾(n):
The permutation (¾(1) : : : ¾(s) : : : ¾(r) : : : ¾(n)) is obtained from (¾(1) : : : ¾(r) : : : ¾(s) : : : ¾(n))
by interchanging 2 numbers, so its sign is different, and detB = ?detA.
Theorem 1.2 (Determinant of a matrix with 2 equal rows). If 2 rows of a matrix are
equal, then its determinant is equal to 0.
Proof. Suppose rows r and s of matrix A are equal. Interchange them to obtain matrix B.
Then detB = ?detA. On the other hand, B = A, so detB = detA. So, detA = ?detA, and
thus detA = 0.Theorem 1.4. If a row of a matrix A consists entirely of zeros, then detA = 0.
Proof. Let’s multiply the zero row of a matrix A by a nonzero number c to obtain matrix
B. Then detB = c detA, But B = A, so detB = detA, and thus detA = c detA. So,
detA = 0.
Theorem 1.5 (Multilinearity by rows). If in matrix A row ar can be represented as sum
of rows b and c, i.e. arj = bj + cj , i.e.Theorem 1.7 (Determinant of a triangular matrix). The determinant of a triangular
matrix is equal to the product of its diagonal elements.
Proof. The product of diagonal elements is included into he expression for the determinant,
and its sign is “+”. All other terms are equal to 0, if the matrix is triangular. Let’s prove itSo, now we know what happens with the determinant after applying elementary row operations.
So, we can now give the algorithm of computing the determinant.
Algorithm. Transform matrix A by elementary row operation to the triangular form keeping
track of how the determinant changes.terminant by the following way:Theorem 1.4. If a row of a matrix A consists entirely of zeros, then detA = 0.
Proof. Let’s multiply the zero row of a matrix A by a nonzero number c to obtain matrix
B. Then detB = c detA, But B = A, so detB = detA, and thus detA = c detA. So,
detA = 0.
Theorem 1.5 (Multilinearity by rows). If in matrix A row ar can be represented as sum
of rows b and c, i.e. arj = bj + cj , i.e.                                                                       
Theorem 1.6 (3rd elementary row operation). If B is obtained from A by adding a row
r multiplied by c to row s, then detB = detA.                                                       Theorem 1.7 (Determinant of a triangular matrix). The determinant of a triangular
matrix is equal to the product of its diagonal elements.
Proof. The product of diagonal elements is included into he expression for the determinant,
and its sign is “+”. All other terms are equal to 0, if the matrix is triangular. Let’s prove it.
3
Let a1k1a2k2 : : : ankn 6= 0. Then
k1 ¸ 1; k2 ¸ 2 : : : ; kn ¸ n
(otherwise the term is equal to 0). But (k1k2 : : : kn) is a permutation of numbers from 1 to n,
so
k1 + k2 + ¢ ¢ ¢ + kn = 1 + 2 + ¢ ¢ ¢ + n;
and it is possible only if
k1 = 1; k2 = 2; : : : ; kn = n:
So, now we know what happens with the determinant after applying elementary row operations