Lect10 - Chapter 2. Determinants Math1111 Determinants...

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Unformatted text preview: Chapter 2. Determinants Math1111 Determinants Basic Properties Theorem 2.1.3 Let A be an n × n triangular matrix, i.e. A =        a 11 a 12 ··· a 1 n a 22 ··· a 2 n . . . . . . a nn        or A =        a 11 a 21 a 22 . . . . . . a n 1 ··· a nn        Then det A = a 11 a 22 ··· a nn . Chapter 2. Determinants Math1111 Determinants Basic Properties Theorem 2.1.3 Let A be an n × n triangular matrix, i.e. A =        a 11 a 12 ··· a 1 n a 22 ··· a 2 n . . . . . . a nn        or A =        a 11 a 21 a 22 . . . . . . a n 1 ··· a nn        Then det A = a 11 a 22 ··· a nn . Proof. By induction on n and expand along the row with at most one nonzero entry. Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Theorem 2.1.4 Let A be a square matrix. (i) If A has a zero row or zero column, then det A = . (ii) If A has two identical rows or two identical columns, then det A = . Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Theorem 2.1.4 Let A be a square matrix. (i) If A has a zero row or zero column, then det A = . (ii) If A has two identical rows or two identical columns, then det A = . Proof. (i) By Theorem 2.1.1 Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Theorem 2.1.4 Let A be a square matrix. (i) If A has a zero row or zero column, then det A = . (ii) If A has two identical rows or two identical columns, then det A = . Proof. (i) By Theorem 2.1.1 (ii) Exercise. Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Theorem 2.1.2 Let A be a square matrix. Then det ( A T ) = det A . Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Theorem 2.1.2 Let A be a square matrix. Then det ( A T ) = det A . Proof. (By induction on the order of A ) When n = 1 , A T = A . ∴ det A T = det A . Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Theorem 2.1.2 Let A be a square matrix. Then det ( A T ) = det A . Proof. (By induction on the order of A ) When n = 1 , A T = A . ∴ det A T = det A . Assume the statement det A T = det A holds when the order of A is n- 1 . Consider the case of order n . Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Let A = ( a ij ) n × n . Write B = A T and B = ( b ij ) . Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Let A = ( a ij ) n × n . Write B = A T and B = ( b ij ) . Expand along the 1 st row, det A = a 11 A 11 + a 12 A 12 + ··· a 1 n A 1 n . Expand along the 1 st column, det B = b 11 B 11 + b 21 B 21 + ··· b n 1 B n 1 . Chapter 2. Determinants Math1111 Determinants Basic Properties (Cont’d) Let A = ( a ij ) n × n . Write B = A T and B = ( b ij ) ....
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This note was uploaded on 04/30/2010 for the course MATH 1111 taught by Professor Dr,li during the Spring '10 term at HKU.

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Lect10 - Chapter 2. Determinants Math1111 Determinants...

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