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4.5 Characterization

# 4.5 Characterization - 4.5 4.5 A characterization of the...

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4.5. 4.5. A characterization of the Determinants 1. (a) F (b) T (c) T (d) F ( δ ( B ) = - δ ( A )) (e) F ( δ ( I ) = 1) (f) T (p.238, 239) 2. Determine all the 1 - linear functions δ : M 1 × 1 ( F ) F Identity function 3. No Let A = a 1 a 2 a 3 + tv , v = ( b 1 , b 2 , b 3 ) δ ( A ) = k 6 = k + tv = δ a 1 a 2 a 3 + a 1 a 2 v 4. No Let A = a 1 a 2 + kv a 3 , v = ( b 1 , b 2 , b 3 ) δ ( A ) = a 2 6 = a 2 + ka 2 = δ a 1 a 2 a 3 + a 1 v a 3 215 PNU-MATH

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4.5. 5. Yes (i) δ a 1 + kv a 2 a 3 = ( A 11 + kb 1 ) A 23 A 32 = A 11 A 23 A 32 + k ( b 1 A 23 A 32 ) = δ a 1 a 2 a 3 + k v a 2 a 3 (ii) δ a 1 a 2 + kv a 3 = A 11 ( A 23 + kb 3 ) A 32 = A 11 A 23 A 32 + k ( A 11 b 3 A 32 ) = δ a 1 a 2 a 3 + k a 1 v a 3 (iii) δ a 1 a 2 a 3 + kv = A 11 A 23 ( A 32 + kb 2 ) = A 11 A 23 A 32 + k ( A 11 A 23 b 2 ) = δ a 1 a 2 a 3 + k a 1 a 2 v 6. No δ a 1 a 2 a 3 + kv = A 11 + A 23 +( A 32 + kb 2 ) 6 = ( A 11 + A 23 + A 32 )+ k ( A 11 + A 23 + b 2 ) = δ a 1 a 2 a 3 + k a 1 a 2 v 7. Yes (i) δ a 1 + kv a 2 a 3 = ( A 11 + kb 1 ) A 21 A 32 = A 11 A 21 A 32 + k ( b 1 A 21 A 32 ) = δ a 1 a 2 a 3 + 216 PNU-MATH
4.5. k v a 2 a 3 (ii) δ a 1 a 2 + kv a 3 = A 11 ( A 21 + kb 1 ) A 32 = A 11 A 21 A 32 + k ( A 11 b 1 A 32 ) = δ a 1 a 2 a 3 + k a 1 v a 3 (iii) δ a 1 a 2 a 3 + kv = A 11 A 21 ( A 32 + kb 2 ) = A 11 A 21 A 32 + k ( A 11 A 21 b 2 ) = δ a 1 a 2 a 3 + k a

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