3.1 - 3.1 Introduction to Determinants Notation A ij is the...

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3.1 Introduction to Determinants Notation: A ij is the matrix obtained from matrix A by deleting the i th row and j th column of A . EXAMPLE: A ± 1234 5678 9 10 11 12 13 14 15 16 A 23 ± Recall that det ab cd ± ad ± bc and we let det ± a ² ± a . For n ² 2 ,the determinant of an n ² n matrix A ± ± a ij ² is given by det A ± a 11 det A 11 ± a 12 det A 12 ³ ³ ³ ´ ± 1 µ 1 ± n a 1 n det A 1 n ± ± j ² 1 n ´ ± 1 µ 1 ± j a 1 j det A 1 j 1
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EXAMPLE: Compute the determinant of A ± 120 3 ± 12 201 Solution det A ± 1det ± 01 ± 2det 32 21 ² 0det 3 ± 1 20 ± ______________________________ ± ______ Common notation: det ± . So 3 ± ± 1 ± ± 2 ² 0 3 ± 1 The ± i , j ² -cofactor of A is the number C ij where C ij ± ± ± 1 ² i ± j det A ij . 3 ± ± 1 C 11 ² 2 C 12 ² 0 C 13 (cofactor expansion across row 1) 2
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THEOREM 1 The determinant of an n ± n matrix A can be computed by a cofactor expansion across any row or down any column: det A ² a i 1 C i 1 ³ a i 2 C i 2 ³ ± ³ a in C in (expansion across row i ) det A ² a 1 j C 1 j ³ a 2 j C 2 j ³ ± ³ a nj C nj (expansion down column j ) Use a matrix of signs to determine ² ± 1 ³ i ± j ³ ± ³ ± ± ³ ± ± ³ ± ³ ± ´´´µ EXAMPLE: Compute the determinant of A ² 120 3 ± 12 201 using cofactor expansion down column 3. Solution 3 ± ²
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