UNIT18 - Unit 18 Determinants Associated with each square...

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Unit 18 Determinants Associated with each square matrix is a number called its determinant. We defne the determinant and look at some oF its properties in this section. Defnition 19.1. G ivenan( n × n ) matrix A we defne the submatrix A ij oF A as the matrix obtained by deleting the i th row and j th column oF A . Example 1. IF A = 1213 2131 2534 3789 then A 11 = 131 534 789 and A 23 = 123 254 379 Defnition 19.2. The Determinant Defnition-Part 1 Let A =( a ij )bean( n × n ) matrix then the determinant oF A is defned as : (i) det ( A )= a 11 iF n = 1. (i.e. iF A =[ a ]then det ([ a ]) = a ) (ii) det ( A a 11 a 22 - a 12 a 21 iF n = 2. (i.e. det ± ab cd ² = ad - bc ) (iii) det ( A a 11 ( detA 11 ) - a 12 det ( A 12 ) ... +( - 1) 1+ j a 1 j det ( A 1 j )+ ...+( - 1) 1+ n a 1 n det ( A 1 n ) 1
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= n j =1 ( - 1) j +1 a 1 j ( det ( A 1 j )) if n> 2 This process is called expanding the determinant along the frst row of A . Example 2. Suppose A = ± 12 25 ² Then by (ii) above we have: det( A ) = (1)(5) - (2)(2) = 1 Example 3. Suppose A = 123 213 321 Find det ( A ) Solution: Looking at the de±nition of the determinant we apply (iii) to get: det ( A )= ( - 1) 1+1 (1) det ( A 11 )+( - 1) 1+2 (2) det ( A 12 - 1) 1+3 (3) det ( A 13 ) =(1) det ( A 11 ) - (2) det ( A 12 )+(3) det ( A 13 ) det ± 13 21 ² - (2) det ± 23 31 ² +(3) det ± 32 ² =(1)( - 5) - (2)( - 4) + (3)(1) = 6 Suppose we are asked to ±nd the determinant of a matrix such as A = 1234 2131 2000 3789 Then det ( A )=( - 1) 2 (1) det ( A 11 - 1) 3 (2) det ( A 12 - 1) 4 (3) det ( A 13 - 1) 5 (4) det ( A
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UNIT18 - Unit 18 Determinants Associated with each square...

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