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**Unformatted text preview: **Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order AMA1061
FOUNDATION MATHEMATICS
DEPARTMENT OF APPLIED MATHEMATICS
Lecturer: Dr. Catherine LIU
Contact: 2766 6931 (O); Ofﬁce Venue: HJ616
Consultation Hours: 2:45pm-4:45pm, Wednesday 17/09/2012
AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 1 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Lecture 1
MATRICES AND DETERMINANTS References:
Section 9.1 & 9.2 & Chapter 8
Chung, K. C. A Short Course in Calculus and Matrices, 2nd Ed., McGraw Hill AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 2 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Outline 1 2 3 4 5 Introduction to Matrices
Determinants
Basic properties of determinants
Cramer’s rule for linear equations
Determinants of higher order. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 3 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Deﬁnition of Matrix
A matrix is a rectangular array of scalars of the form
⎡
⎤
a11 a12 ⋅ ⋅ ⋅ a1n
⎢ a21 a22 ⋅ ⋅ ⋅ a2n ⎥
⎢
⎥
⎢.
.
.⎥
.
.⎦
⎣.
.
.
.
am1 am2 ⋅ ⋅ ⋅ amn (1) This matrix is called an m × n matrix . It consists of m rows and n columns,
the i -th row (or simply row i ) being the 1 × n matrix [ai 1 ai 2 ⋅ ⋅ ⋅ ain ] and
the j -th column (or column j ) the m × 1 matrix
⎡
⎤
a1j
⎢ a2j ⎥
⎢
⎥
⎢ . ⎥.
.⎦
⎣.
anj Remark: The entry aij which belongs to row i and column j is called the (i , j )-entry of the matrix. The matrix (1) can be written in abbreviated forms such as [aij ]m×n or [aij ] or [A] or simply A. The 1 × 1 matrix [ ] is just the scalar
AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants . 17/09/2012 4 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Deﬁnition of Matrix Example
The following are examples of matrices. A= 4.5 1.4
2.3 5.9 , ⎡ ⎤
5
1
B = ⎣ 2 −9 ⎦ ,
3
2 ⎡⎤
2
c = ⎣6⎦ .
5 The ﬁrst, denoted by A, is a 2 × 2 matrix. The second, denoted by B , is a 3 × 2
matrix. The third, denoted by c is a 3 × 1 matrix. If the entries of the matrix A
above are denoted by aij , then a11 = 4.5, a12 = 1.4, a21 = 2.3, a22 = 5.9. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 5 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Transposition
If A = [aij ] is m × n, then its transpose is the n × m matrix, denoted by AT ,
obtained by interchanging the rows and columns of A, i.e.
AT = [bij ] where bij = aji for all i = 1, . . . , n; j = 1, . . . , m. Example
If A, B and c are the matrices given in the previous example, then
AT = 4.5 2.3
1.4 5.9 AMA1061 (By Catherine Liu) , BT = 5
23
1 −9 2 , Lecture 1 Matrices and Determinants cT = [ 2 6 5 ]. 17/09/2012 6 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Vectors An m × 1 matrix is called a column m-vector or an m-vector or a
column-vector or simply a vector . Column-vectors are denoted by bold-faced
lower-case letters like a, x in printed form and is hand-written as a , x , etc. A
∼∼ 1 × n matrix is called a row-vector . As row-vectors are transposes of
column-vectors, they are denoted by aT , xT , etc. and are written as a T , x T ,
∼∼
etc. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 7 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Equality of matrices Two m × n matrices A and B are said to be equal ,
written A = B , if aij = bij for all i = 1, . . . , m;
j = 1, . . . , n.
Example If E = 42
15 and F = 41
, then E ∕= F but
25 E = FT. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 8 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Addition and subtraction
We can add two m × n matrices. The sum, denoted by A + B is an m × n
matrix C = [cij ] such that cij = aij + bij for all i , j .
Similarly, the difference, denoted by A − B is an m × n matrix D = [dij ] such
that dij = aij − bij for all i , j . Example
42
35 + −2 1
2 −2 = 4−2 2+1
3+2 5−2 = 2
5 3
3 . 42
35 − −2 1
2 −2 = 4+2 2−1
3−2 5+2 = 6
1 1
7 . AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 9 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Scalar multiplication If k is a scalar (real or complex), then kA is the matrix C = [cij ] such that
cij = kaij for all i , j . Example
3 4
2
3 −5 = AMA1061 (By Catherine Liu) 12
6
9 −15 , k −2 1
34 Lecture 1 Matrices and Determinants = −2k
3k k
4k 17/09/2012 . 10 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Zero matrix
This is a matrix all whose entries are 0. Zero matrices
are denoted by O . However, zero vectors are usually
denoted by 0.
Example O= 00
00 AMA1061 (By Catherine Liu) , ⎡ ⎤
000
O = ⎣ 0 0 0 ⎦,
000 Lecture 1 Matrices and Determinants ⎡⎤
0
0 = ⎣0⎦ .
0 17/09/2012 11 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Multiplication of vectors
If a row n-vector is multiplied by a column n-vector on the right, then the
product is a scalar which is equal to the sum of the products of the
corresponding entries in the two vectors. Using symbols, if
⎡⎤
b1
⎢b2 ⎥
⎢⎥
aT = [a1 a2 ⋅ ⋅ ⋅ an ] and b = ⎢ . ⎥
⎣.⎦
.
bn
then aT b = a1 b1 + a2 b2 + ⋅ ⋅ ⋅ + an bn . Example
3 −2 1 AMA1061 (By Catherine Liu) ⎡ ⎤
2
⎣ 4 ⎦ = 3 × 2 + (−2) × 4 + 1 × (−3) = −5.
−3
Lecture 1 Matrices and Determinants 17/09/2012 12 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Multiplication of matrices If A = [aij ] is m × n, B = [bij ] is n × p then the product AB is the m × p matrix
C = [cij ] such that
cij = ai 1 b1j + ai 2 b2j + ⋅ ⋅ ⋅ + ain bnj
for all i , j , i.e. the entry cij is equal to row i of A right-multiplied by column j of
B . The product AB is well-deﬁned only when the number of columns of A is
equal to the number of rows of B . AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 13 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Example
If A = 5
2 1
2 2
−1 3
3 ⎡ 1
⎢ −2
and B = ⎢
⎣
3
−4 2
3
−3
2 ⎤
3
−1 ⎥
⎥ then
−1 ⎦
2 c11
c21 c12
c22 c13
c23 AB = = −3
−17 13
19 18
11 where the entries cij are computed as follows:
c11 = 5 × 1 + 1 × ( −2 ) + 2 × 3 + 3 × ( −4 ) = −3 . c12 = 5 × 2 + 1 × 3 + 2 × (−3) + 3 × 2 = 13. c13 = 5 × 3 + 1 × (−1) + 2 × (−1) + 3 × 2 = 18. c21 = 2 × 1 + 2 × (−2) + (−1) × 3 + 3 × (−4) = −17. c22 = 2 × 2 + 2 × 3 + (−1) × (−3) + 3 × 2 = 19. c23 = 2 × 3 + 2 × (−1) + (−1) × (−1) + 3 × 2 = 11. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 14 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Properties
If the sums are well-deﬁned, we have:
1 A+B =B+A 2 (A + B ) + C = A + (B + C ) 3 A+O =A 4 A + (−A) = O where −A = (−1)A TT 5 (A ) = A 6 (A + B )T = AT + B T If the products are well-deﬁned, then
7 A(BC ) = (AB )C 8 A(B + C ) = AB + AC , (A + B )C = AC + BC 9 (AB )T = B T AT 10 There are examples of A and B for which AB ∕= BA. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 15 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Square matrices
An n × n matrix is a square matrix . Its order is n.
Note that a square matrix of order 1 is just a number.
Example The following matrices A, B , C are square matrices. A= 52
13 AMA1061 (By Catherine Liu) ⎡ ⎤
⎡
⎤
203
200
, B = ⎣ 4 1 0 ⎦, C = ⎣ 0 1 0 ⎦
724
004
Lecture 1 Matrices and Determinants 17/09/2012 16 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Diagonal elements If A = [aij ] is a square matrix of order n, then the
entries a11 , a22 , . . . , ann are the diagonal elements of
A. For example, the diagonal elements of the matrix
⎡
⎤
2 5 −3
A = ⎣ 3 −1 7 ⎦
204
are 2, −1, 4. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 17 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Diagonal matrix
A diagonal matrix is a square matrix whose entries are all zero except
possibly the diagonal elements. Example
The matrices
⎡ ⎤
200
A = ⎣ 0 −1 0 ⎦
004 are diagonal matrices. ⎡ ⎤
200
and B = ⎣ 0 1 0 ⎦
000 We use diag(a11 , . . . , ann ) to denote the diagonal matrix of order n whose
diagonal elements are a11 , . . . , ann . For example, the matrix B above can be
represented by diag(2, 1, 0). AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 18 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Identity matrix
A diagonal matrix whose diagonal elements are all unity is called an identity
matrix . It is denoted by In if the speciﬁcation of its order n is necessary.
Usually we denote it simply by I . Example
The matrices
I2 =
are identity matrices. 10
01 , ⎡ ⎤
100
I3 = ⎣ 0 1 0 ⎦ ,
001 Theorem
If A is a square matrix of order n and I the identity matrix of the same order
then AI = IA = A. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 19 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Determinant
Using matrix terminology, a determinant is deﬁned for every square matrix. If
A is a square matrix, then a determinant is formed and is called the
determinant of A and is denoted by det A or det(A).
For example, let
⎤
⎡
3
2 −1
0 ⎦.
A = ⎣ 2 −1
(2)
2
3
1
Then det A = AMA1061 (By Catherine Liu) 3
2 −1
2 −1
0
2
3
1 = −15. Lecture 1 Matrices and Determinants (3) 17/09/2012 20 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Second-order determinants
By a determinant of the second order we mean the symbol
a1 b1 a2 b2 which represents the number a1 b2 − a2 b1 evaluated in the following way:
a1 b1 a2 b2 The second order determinant has two (horizontal) rows and two (vertical)
columns1 . 1 Afterward Ri is for row i and Cj for column j in this course. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 21 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Cramer’s rule
Consider a system of two linear equations in two unknowns x and y :
a 1 x + b 1 y = k1
a 2 x + b 2 y = k2 (4) An ordered pair of numbers (x , y ) is said to be a solution of the system if x and y satisfy the equations in the system. By some
calculation, if a1 b2 − a2 b1 ∕= 0, the solution is given by x= k1 b 2 − k2 b 1 , y= , a1 b2 − a2 b1 a 1 k2 − a 2 k1 y= a1 b2 − a2 b1 (5) . With the notation of determinants, we may now write (5) as x= k1
k2 b1
b2 ÷ a1
a2 b1 k1 a2 b2 a1 k2 ÷ a1 b1 a2 b2 (6) . The formulas (6) are called Cramer’s rule in which we assume that the denominator
a1
a2 AMA1061 (By Catherine Liu) b1
b2 ∕= 0. Lecture 1 Matrices and Determinants 17/09/2012 22 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Examples
Example
Solve the following linear system by Cramer’s rule.
2x +3y =4
4x + y = −2 Solution.
Using Cramer’s rule (6), we get
x= 4 3 −2 1 ÷ 2 3 4 1 AMA1061 (By Catherine Liu) = 10
= −1,
−10 y= 2 4 4 −2 Lecture 1 Matrices and Determinants ÷ (−10) = −20
= 2.
−10 17/09/2012 23 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Third-order determinants
A third-order determinant consists of three rows and three columns. It is
denoted by
a1 b1 c1
a2 b2 c2 a3 b3 c3 and is a number which can be found as follows:
a1 b1 c1 a2 b2 c2 a3 b3 c3 b2 c2 = a1 a2 c2 b3 c3 a2 b2 a3 b3 + c1 − b1
a3 c3 (7)
= a1 (b2 c3 − b3 c2 ) − b1 (a2 c3 − a3 c2 ) + c1 (a2 b3 − a3 b2 )
= a1 b2 c3 + a3 b1 c2 + a2 b3 c1 − a3 b2 c1 − a1 b3 c2 − a2 b1 c3 .
AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants (8)
(9)
17/09/2012 24 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order First row expansion
Formula (7) or (8) is the result obtained by expanding the determinant along
the ﬁrst row . In this method, in order to ﬁnd the coefﬁcient of b1 , say, we
imagine the row and column containing b1 erased and the determinant of the
remaining entries put down as they stand. Also we put the + sign to a1 (the
upper-left entry), − sign to b1 (the one adjacent to a1 ), etc. so that the signs
associated with the entries of the determinant appear alternately as shown
below. Thus we have +a1 , −b1 , +c1 , −c2 , +b2 , etc. with
+ − + − + − + − + AMA1061 (By Catherine Liu) a1 c1 a2 b2 c2 . a3 overlaying on b1 b3 c3 Lecture 1 Matrices and Determinants 17/09/2012 25 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Row and Column expansion
We can also get the same value of the determinant by expanding along any
row or column. For example, if we expand along the second column, we get
a1 b1 c1 a2 b2 c2 a3 b3 c3 a2 c2 = −b1 a1 c1 + b2
a3 c3 a1 c1 a2 c2 − b3
a3 c3 = −b1 (a2 c3 − a3 c2 ) + b2 (a1 c3 − a3 c1 ) − b3 (a1 c2 − a2 c1 )
= a1 b2 c3 + a3 b1 c2 + a2 b3 c1 − a3 b2 c1 − a1 b3 c2 − a2 b1 c3 .
which is the same as (9). It can be proved that all values obtained by
expansion along a row or column are equal. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 26 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Sarrus’ rule for 3rd order determinant:
This rule says that we evaluated the determinant directly as
a1 b1 c1 a2 b2 c2 a3 b3 c3 = a1 b2 c3 + b1 c2 a3 + c1 a2 b3 − a3 b2 c1 − b3 c2 a1 − c3 a2 b1 . The rule can be easily memorized using the following diagram. The six terms
on the RHS are obtained by multiplying the entries following the arrows. A
downward right arrow is associated with a positive sign while an upward right
arrow a negative sign.
a1 c1 a1 b1 a2 b2 c2 a2 b2 a3 AMA1061 (By Catherine Liu) b1 b3 c3 a3 b3 Lecture 1 Matrices and Determinants 17/09/2012 27 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Examples
Example
Evaluate the determinant
3 2 2 3 4 5 D= 1 3 1 by
(a) expanding along the ﬁrst row; (b) expanding along the second column; (c) expanding along the third row; (d) Sarrus’ rule. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 28 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Solution
3 4 3 (a) 1 2 4 5 D=3 1 2 3 5 3 +2 −1 (expansion along R1 ) = 3(−9) − (−18) + 2(−9) = −27.
2
(b) 4 D = −1 3 2 +3
5 3 2 2 4 −3 1 5 1 (expansion along C2 ) = −(−18) + 3(−7) − 3(8) = −27.
1
(c) 2 D=5 3 2 3 3 1 2 3 +1 −3
4 2 4 (expansion along R3 ) = 5(−2) − 3(8) + (7) = −27.
(d) By Sarrus’ rule
D = (3)(3)(1) + (1)(4)(5) + (2)(2)(3) − (5)(3)(2) − (3)(4)(3) − (1)(2)(1)
= −27. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 29 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Basic properties
1. The value of a determinant is not changed if we put its rows as columns,
in the same consecutive order, e.g.
a1 b1 c1 a1 a2 a3 a2 b2 c2 = b1 b2 b3 a3 b3 c3 c1 c2 c3 (Transposition) 2. Interchanging any two columns (or any two rows) changes the sign of
the determinant, e.g.
a1 b1 c1 c1 b1 a1 a2 b2 c2 = − c2 b2 a2 a3 b3 c3 b3 a3 c3 (C1 ∼ C3 ) where the ﬁrst and third columns (C1 , C3 ) of the ﬁrst determinant have
been interchanged to form the second determinant.
AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 30 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order 3. If a determinant has two rows (or two columns) identical, its value is
zero, e.g.
a1 b1 c1
a2 b2 c2 = 0 a2 b2 (R2 = R3 ) c2 since the second and third rows (R2 , R3 ) are identical.
4. A common factor of any row or column can be taken out to multiply the
remaining determinant value, e.g.
kma1 mc1 ka2 b2 ka3 AMA1061 (By Catherine Liu) mb1 a1 b1 c1 c2 = km a2 b2 c2 . b3 c3 a3 b3 c3 Lecture 1 Matrices and Determinants 17/09/2012 31 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order 5. If each element of a row (or a column) is the sum of two or more terms,
the determinant can be expressed as the sum of two or more
determinants, e.g.
a1 + m1 b1 c1 a1 b1 c1 m1 b1 c1 a2 + m2 b2 c2 = a2 b2 c2 + m2 b2 c2 . a3 + m3 b3 c3 b3 c3 m3 b3 c3 a3 A more general formula is
ka1 + lm1 b1 c1 a1 b1 c1 m1 b1 c1 ka2 + lm2 b2 c2 = k a2 b2 c2 + l m2 b2 c2 . ka3 + lm3 b3 c3 a3 b3 c3 b3 c3 AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants m3 17/09/2012 (10) 32 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order 6. By choosing l = −a3 /b3 (possible if b3 ∕= 0) so that a3 + lb3 = 0, we can
simplify the determinant as
a1 b1 c1 a1 + lb1 b1 c1 a2 b2 c2 = a2 + lb2 b2 c2 . a3 b3 c3 b3 c3 0 (C1 + l C2 → C1 ) Note that in the result, a3 is replaced by 0 since a3 + lb3 = 0. This result
follows by putting k = 1 and mi = bi in (10). This helps to change one or
more entries of the determinant to 0 without changing the value of the
determinant. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 33 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Remark
In No. 6 above, we used the abbreviation C1 + l C2 → C1 to represent
the column operation:
New column 1 is formed by adding old column 1 to l times
column 2
For convenience, we will simply write C1 + l C2 to represent this
operation in the future. The convention being used is: when we write
k Ci + mCj (i ∕= j ) to represent a column operation, the result is used to
replace Ci , the column being ﬁrst written.
Row operations k Ri + mRj (i ∕= j ) are deﬁned similarly.
If a determinant D is transformed to another determinant E by the
operation: k Ci + mCj or k Ri + mRj (i ∕= j ), then E = kD . This fact follows
from (10) above. AMA1061 (By Catherine Liu) Lecture 1 Matrices and Determinants 17/09/2012 34 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Example
Example
2 3 23 1 6 16 3 Evaluate the determinant Δ = 8 38 . Solution 1. By expansion along a row or a column, or using Sarrus’ rule, we can get Δ = 0. For details, see Example 14 on
page 28.
Solution 2.
2
= 3
6 16 8 0 38 = −9 −9 1 6 16 0 23 1
3 Δ −10 0 −10 90 1 1 1 6 16 0 = 1 1 = 0. Solution 3. Obviously the third column is 10 times the ﬁrst plus the second. Therefore
2
= 3
6 16 8 38 2 AMA1061 (By Catherine Liu) = 10 3
6 1 8 3 2 2 1
3 23 1
3 Δ + 3 3 1 6 6 3 8 8 Lecture 1 Matrices and Determinants = 10 × 0 + 0 = 0. 17/09/2012 35 / 46 Outline Introduction Determinants Basic Properties Cramer’s rule for linear equations Determinants of higher order Example
1
1
1 Factorize the determinant Δ = a3
b3
c3 a
b
...

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