4.2 Determinants of Order N

4.2 Determinants of Order N - 4.2 4.2 Determinants of Order...

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4.2. 4.2. Determinants of Order n 1. (a) F (b) T (Theorem 4.4) (c) T (Corollary to Theorem 4.4) (d) T (Theorem 4.5) (e) F (det( B ) = k det( A )) (f) F (det( B ) = det( A )) (g) F (If A M n × n ( F ) and rankA = n A : invertible det A 6 = 0) (h) T 2. k = 3 3 3. k = 42 4. k = 2 5. det( A ) = - 12 6. det( A ) = - 13 192 PNU-MATH

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4.2. 7. det( A ) = - 12 8. det( A ) = - 13 9. det( A ) = 22 10. det( A ) = 4 + 2 i 11. det( A ) = - 3 12. det( A ) = 154 13. det( A ) = - 8 14. det( A ) = - 168 15. det( A ) = 0 16. det( A ) = 36 17. det( A ) = - 49 193 PNU-MATH
4.2. 18. det( A ) = 10 19. det( A ) = - 28 - i 20. det( A ) = 17 - 3 i 21. det( A ) = 95 22. det( A ) = 100 23. Let A = a 11 a 12 · · · a 1 n a 22 · · · a 2 n . . . . . . a nn By expanding along the first column, we have det A = a 11 det a 22 a 23 · · · a 2 n a 33 · · · a 3 n . . . . . . a nn = a 11 a 22 det a 33 a 34 · · · a 3 n a 44 · · · a 4 n . . . . . . a nn . . . = a 11 a 22 · · · a nn 194 PNU-MATH

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4.2. 24. If A has a row consisting entirely of zeros, then det( A ) = 0 Let i - th row of A is the zero row Since det( A ) = n j =1 ( - 1) i + j A ij det( ˜ A ij ) and A ij = 0 , ( j = 1 , 2 , · · · , n ) det( A ) = 0 25.
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