Lect10 - Chapter 2 Determinants Math1111 Determinants Basic Properties Theorem 2.1.3 a11 A= a12 a22 Let A be an n n triangular matrix i.e a1n a2n ann

# Lect10 - Chapter 2 Determinants Math1111 Determinants Basic...

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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 = 0 . (ii) If A has two identical rows or two identical columns, then det A = 0 .
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 = 0 . (ii) If A has two identical rows or two identical columns, then det A = 0 . 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 = 0 . (ii) If A has two identical rows or two identical columns, then det A = 0 . 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 2.1.2LetAbe a square matrix. Thendet(AT) =detA.Proof. (By induction on the order ofA)Whenn=1,AT=A.detAT=detA.Assume the statementdetAT=detAholds when the order ofAisn-1.Consider the case of ordern.
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 ) . 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 . As B = A T , b j 1 = a 1 j . Remains to show B j 1 = A 1 j . B j 1 = ( - 1 ) j + 1 det (( j , 1 ) th minor of A T )
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 .