MAT1341-L16.Determinantes-VC

Along a row or column is more efficient when it is

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along a row or column is more efficient when it is applied to the row or column with the largest number of zeros. Example 12. det 1 5 6 0 13 11 0 40 26 = ( 1) [( 13)(26) ( 11)(40)] = 102 . Properties of Determinants The following result summarizes the basic properties of determinants. Theorem 13. Let A be a square matrix. (1) If A has a zero row or column, then det( A ) = 0 . (2) If A has two proportional rows (or two proportional columns), then det( A ) = 0 . (3) If A is a upper (lower) triangular matrix, then det( A ) is the product of its diagonal components. (4) If A is a diagonal matrix, then det( A ) is the product of its diag- onal components. (5) det( I n ) = 1 . (6) det( A ) = det( A T ) . Example 14. Some examples are: det 7 4 0 1 1 3 0 4 9 2 0 9 7 1 0 10 = 0 , det 7 4 4 1 0 3 8 7 0 0 2 4 0 0 0 10 = 420 5
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Theorem 15. Let A be an n × n matrix. (1) If B is the matrix obtained by interchanging any two rows of A , then det( B ) = det( A ) (2) If B is the matrix obtained by multiplying a single row of A by any non-zero scalar α , then det( B ) = α det( A ) (3) If B is the matrix obtained by adding a multiple of one row of A to another row of A , then det( B ) = det( A ) Caution: Notice that in (2) from the theorem above the number α is multiplying only a single row . This implies that for an n × n matrix A , regarding αA as the scalar product for matrices by a non-zero scalar α , we get det( α A ) = α n det( A ) . Remark 16 . Since det( A ) = det( A T ), we can replace the word row in the previous theorem and get similar statements for the respective column operations . Example 17. If det a b c d e f g h i = 10, then det f e d c b a i h g = 10 , det b a 2 c e d 2 f h g 2 i = 20 , det a d 2 a 5 g + a b e 2 b 5 h + b c f 2 c 5 i + c = 50 , det c b a 2 f 2 e 2 d i h g = 20 6
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Computational Techniques Method # 1: Using elementary row operations transform any ma-
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