UNIT19 - Unit 19 Properties of Determinants Theorem 20.1....

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Unit 19 Properties of Determinants Theorem 20.1. Suppose A 1 and A 2 are identical n × n matrices with the exception that one row ( or column ) of A 2 is obtained by multiplying the corresponding row ( or column ) of A 1 by some nonzero constant c .Th en det ( A 2 )= c ( det ( A 1 )) This theorem may be used in a variety of situations to make a seemingly difficult determinant ( due to large numbers or fractions involved ) into a much easier one, as the following examples demonstrate. Example 1. It is easily seen that det 293 40 0 31 1 =-48 Therefore det 693 12 0 0 91 1 =3( 48) = 144 Example 2. Find detA where A = 2 04 1 62 0 39 1 2 Solution: det 1 1 2 =2 × det 10 2 1 1 2 =3 × 2 × det 2 1 13 4 × 3 × 2 × det 2 1 22 11 4 × 3 × 3 × 2 × det 1 1 21 2 (expanding along the ±rst row we get) =36 { 1[ 4 1] 0[2 1]+1[1+2] } = 36( 2)= 72 1
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Example 3. Find the determinant of 1 3 0 3 4 2 5 1 3 2 1 8 - 3 4 5 4 Solution: By taking a factor of 1 12 from row 1, a factor of 1 10 from row 2, and one of 1 8 from row 3, we see that det 1 3 0 3 4 2 5 1 3 2 1 8 - 3 4 5 4 = 1 12 × 1 10 × 1 8 × det 40 9 4 10 15 16 1 0 = 1 12 × 1 10 × 1 8 × ( 2) × det 40 9 451 5 131 0 = 1 480 × 83 = 83 480 Remark. Be warned that it is not true in general that if c is a constant then det ( cA )= cdet ( A ). Since cA is formed by multiplying every entry and hence every row of A by c we may take out a factor of c from each row, so for any n × n matrix A : det ( cA c n det ( A ) So, for example if A is a 3 × 3 matrix, then det (2 A )=2 3 det ( A )=8 det ( A ). Theorem 20.2. More Facts About Determinants Suppose A and B are square n × n matrices, then: 1. If A has one row (column) that is a scalar multiple of another row (column) of A then det ( A )=0 . 2. If two rows ( columns ) of A are interchanged then the determinant changes by a factor of -1. (i.e. the sign is reversed).
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This note was uploaded on 04/15/2010 for the course MATH 20C taught by Professor Lit during the Spring '10 term at UCLA.

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UNIT19 - Unit 19 Properties of Determinants Theorem 20.1....

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