301 - CHAPTER 3 DETERMINANTS Definition. Given any A =...

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1 CHAPTER 3 DETERMINANTS Definition. Given any A R n × n , the determinant of A , denoted by det A or | A | , is defined as det A = a 11 for n = 1 and det A = a 11 ·det A 11 a 12 A 12 + " + ( 1) 1+ n a 1 n A 1 n for n > 1. The ( i , j )-cofactor c ij of A is defined as ( 1) i + j A ij . Definition. Given any A = [ a ij ] R n × n , let A ij be the ( n 1) × ( n 1) submatrix of A with the i th row and j th column of A deleted. i th row j th column Example: bc ad A b A a A c A d A d c b a A = = = = = 12 11 12 11 det det det ] [ and , ] [ A is not invertible [ ac ] T and [ bd ] T are L.D. a = c = 0 or [ ] T = k [ ] T for some k det A = ad bc = 0 Later it will be shown that for any A R n × n , A is not invertible if and only if det A = 0. Example: Find the scalars c for which A cI 2 is not invertible, where Solution: for c = 1 and 3, A cI 2 is not invertible.
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2 Example: + + = + + + 32 31 22 21 13 3 1 33 31
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301 - CHAPTER 3 DETERMINANTS Definition. Given any A =...

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