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KALKULUS DAN ALJABAR LINEARKALKULUS DAN ALJABAR LINEARKALKULUS DAN ALJABAR LINEARKALKULUS DAN ALJABAR LINEARKALKULUS DAN ALJABAR LINEARKALKULUS DAN ALJABAR LINEARKALKULUS DAN ALJABAR LINEARKALKULUS DAN ALJABAR LINEAR
DeterminantDeterminantDeterminantDeterminant
TIM KALIN
• Setelah menyelesaikan pertemuan ini mahasiswa diharapkan :
– Dapat menghitung determinan
– Dapat menyelesaikan Sistem Persamaan Linier dengan
menggunakan determinan
Tujuan Pembelajaran
Page 2Surabaya, 03 Oktober 2012 KALKULUS DAN ALJABAR LINEAR – DETERMINAN MATRIKS
Definition of
Evaluating Evaluating
Determinant Properties of
Determinant
A A
Combinatorial
Outline
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Definition of
Determinant
Determinant
by Row
Reduction
Properties of
Determinant
Function
Combinatorial
Approach To
Determinants
DEFINITION OF DETERMINANT
Determinant of a 2x2 Matrix
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• Determinant : the difference of the products of the two
diagonals of the matrix
• Note : the order is important
Example 1
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Example 1 (Cont.)
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Minor and Cofactors of A Matrix
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Minor and Cofactors of A Matrix (Cont.)
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Minor and Cofactors of A Matrix (Cont.)
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• Find all the minors and cofactors of
Example 2
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Example 2 (Cont.)
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Definition of Determinant
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Cofactor Expansion
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Cofactor expansion along
the i-th rowCofactor expansion along
the j-th row
1. Evaluate det(A) by cofactor expansion along the first row of A.
2. Evaluate det(A) by cofactor expansion along the first column of
Example 3
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2. Evaluate det(A) by cofactor expansion along the first column of
A.
• Soal 1 (ekspansi kofaktor baris)
Example 3 (Cont.)
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• Soal 2 (ekspansi kofaktor kolom)
• Find det(A)
Example 4
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• Solusi
Example 4 (Cont.)
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Adjoint of Matrix A
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• Let
Example 5
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• find the inverse of the matrix A
Example 5 (Cont.)
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Example 5 (Cont.)
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Determinant of Triangular Matrix
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Example 5
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Cramer’s Rule
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• Use Cramer's rule to solve
Example 6
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Example 6 (Cont.)
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EVALUATING DETERMINANT BY
ROW REDUCTION
• Let A be a square matrix. If A has a row of zeros or a column of
zeros, then det(A)=0.
• Let A be a square matrix. Then det(A)=det(AT).
Basic Theorem
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Theorem
Elementary Row Operations and Determinants
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Example 7
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Theorem
Determinant of Elementary Matrices
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Example 8
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• If A is a square matrix with two proportional rows or two
proportional columns, then det(A)=0.
Introducing Zero Rows (Theorem)
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• Manakah yang termasuk matriks proporsional?
Example 9
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Zero Determinant
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Steps :
1. Reduce the given matrix to upper triangular form by
elementary row operations,
2. then compute the determinant of the upper triangular matrix
(an easy computation), and
Evaluating Determinant By Row Reduction
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(an easy computation), and
3. then relate that determinant to that of the original matrix.
• Evaluate det(A) where
Example 10
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Example 10 (Cont.)
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• Find the determinant of
Example 11
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Example 11 (Cont.)
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Steps :
1. Reduce the given matrix to upper triangular form by
elementary column operations,
2. then compute the determinant of the upper triangular matrix
(an easy computation), and
Evaluating Determinant By Column Reduction
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(an easy computation), and
3. then relate that determinant to that of the original matrix.
Compute the determinant of
Example 12
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Example 12 (Cont.)
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• Evaluate det(A) where
Example 13
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Solution
By adding suitable multiples of the second row to the remaining
rows, we obtain
Example 13 (Cont.)
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Example 14 (Cont.)
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Solution
Example 14 (Cont.)
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A
COMBINATORIAL
APPROACH TO
DETERMINANTS
Next Week
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PROPERTIES OF
DETERMINANT
FUNCTION
DETERMINANTS
