1. CHAPTER 2 – DETERMINANTS 2.1. The Determinant Function 2.2. Evaluating Determinants by Row Reduction 2.3. Properties of the Determinant Function 2.4. Cofactor Expansion; Cramer’s Rule

3. Example 1 A = 3 1 4 –2 det(A) = 3  (-2) – 1  4 = –10

4. Example 2 B = 1 2 3 1 2 3 -4 5 6 -4 5 6 7 -8 9 7 -8 9 det(B) = (45+84+96) – (105+(-48)+(-72)) = 240

5. Example 2 B = 1 2 3 – 4 5 6 7 –8 9 det (B) = 1 5 6 – 2 –4 6 + 3 –4 5 – 8 9 7 9 7 –8 = 1  (45 + 48) – 2  (– 36 – 42) + 3  (32 – 35) = 1  ( 93 ) – 2  ( – 78 ) + 3  ( – 3 ) = 93 + 156 – 9 = 240

7. Theorem: If A (n  n) is a triangular / diagonal matrix, then det(A) is the product of the diagonal entries example: A = 2 7 -3 det(A) = 2  (-3)  6 = -36 0 -3 7 0 0 6 “ Proof”: 2 7 -3 2 7 0 -3 7 0 -3 0 0 6 0 0

10. Example 2 B = 1 2 3 1 2 3 – 4 5 6 -4 5 6 7 –8 9 7 –8 9 det(B) = (45+84+96) – (105+(-48)+(-72)) = 240 B’ = 1 2 3 B’’ = 1 2 3 0 13 18 0 13 18 0 -22 -12 0 0 240/13 det(B’’) = 240 = det(B) ERO 3: row 2 + 4  row 1; row 3 – 7  row 1 ERO 3: row 3 + (22/13)  row 2 Note: this “algorithm” is called row reduction

11. Theorem: If A is a square matrix A with proportional rows/columns, then det(A) = 0 Example : A = 2 3 5 2 3 5 4 6 10 0 0 0 8 7 11 8 7 11 det(A) = 0 ERO 3: row 2 – 2  row 1

12. Cramer’s Rule: The solution of a system of linear equation Ax = b where det(A)  0 is unique and is determined by: x j = i = 1, 2, 3, …, n det(A j ) det(A) A j is obtained by replacing the j-th column of matrix A by b

16. Pelajari sendiri semua definisi, teorema, algoritma yang tidak dibahas di kelas Latihan: 2.1. no. 19 2.2. no. 8 2.3. no. 3 2.4. no. 17