ch02 - Chapter 2 Determinants Section 2.1 Exercise Set 2.1...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
44 Chapter 2 Determinants Section 2.1 Exercise Set 2.1 1. 11 71 28 1 29 14 () M == = 11 11 11 12 9 CM + =− = 12 61 24 3 21 34 M = 12 12 1 + 13 67 62 1 2 7 31 M = 13 13 13 7 + = 21 23 83 1 1 M = 21 21 21 1 + = 22 49 1 3 M = 22 22 22 3 + = 23 16 5 M = 23 23 15 + = 31 1 1 9 M = 31 31 9 + 32 8 1 9 M = 32 32 32 9 + = 33 2 1 9 M = 33 33 33 9 + = 3. (a) 13 00 3 41 411 4 3 3 44 0 41 2 M = = 13 13 10 + = (b) 23 416 1 4 412 4 4 6 42 41 42 14 8 56 64 4 96 ( ) M = + =−+ −+− 23 23 19 6 + = (c) 22 41 6 401 4 43 2 16 46 4 42 43 4 2 18 14 12 4 48 M = + 22 22 8 + (d) 21 11 6 101 4 13 2 1 4 8 1 431 72 M = −− + − − = 21 21 17 2 + 5. 35 12 10 22 24 − = 5 2 1 11 22 3 1 11 22 1 4 5 2 3 22 ⎡⎤ ⎢⎥ ⎣⎦ 7. 57 10 49 59 72 7 2 1 59 59 75 59 59 1 27 59 ⎡⎤⎡⎤ ⎢⎥⎢⎥ ⎣⎦⎣⎦
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
SSM: Elementary Linear Algebra Section 2.1 45 9. 2 2 35 32 5 3 56 1 5 52 1 () ( ) ( ) a aa a =− −− − −− =−+ + 11. 21 4 21 421 35 7 35 735 16 2 16 216 252 1 71 436 451 2 76 132 20 7 72 20 84 6 65 [( )( )( ) ( )( )( ) ( )( )( )] [( )( )( ) ( )( )( ) ( )( )( )] [] [ ] −= + + +− − + = +−+ + 13. 300300 30 215215 21 19 419 4 19 3 1 4 051 029 0 11 359 02 4 12 135 123 [( )( )( ) ( )( )( ) ( )( )( )] [ ( )( ) ( )( )( ) ( )( )( )] + + −+ + 15. 2 2 245 28 5 23 13 det( ) ( )( ) ( ) ( ) A =λ− λ+ + λ− + det( A ) = 0 for λ = 1 or 3. 17. det( A ) = ( λ 1)( λ + 1) 0 = ( λ 1)( λ + 1) det( A ) = 0 for λ = 1 or 1. 19. (a) 300 15 25 21 2153 0 0 94 14 19 19 4 34 45 123 + (b) 15 00 0 0 2 1 94 94 1 5 20 0 0 0 123 ( ) + =−−− + (c) 00 30 3 0 215 2 1 5 94 14 1 9 12 0 527 0 12 135 123 ( ) ( ) + = − −− −−−
Background image of page 2
Chapter 2: Determinants SSM: Elementary Linear Algebra 46 (d) 300 25 30 3 0 215 0 1 9 14 14 25 19 4 12 0 9 15 0 12 135 123 () ( ) −= + −− =− − =− (e) 00 3 0 2151 9 4 15 2 1 00 9 1 50 430 135 12 123 ( ) ( ) + =−− −−− + (f) 21 3 0 2150 5 4 19 19 527 0 4 3 0 135 12 123 + − − + 21. Expand along the second column. 37 55 1 5 7 4 0 det( ) [ ( )] A == = 23. Expand along the first column. 222 33 111 0 det( ) kk A kk kk kk =−+ =−−−+− = 25. Expand along the third column, then along the first rows of the 3 × 3 matrices. 33 5 33 5 32 2 2 32 2 2 21 0 2 4 1 0 22 22 22 22 22 2 2 3 5 3 5 10 2 2 2 2 10 1 0 4 0 4 1 3324 38 516 332 5 6 3 128 3 48 240 det( ) [ ()( )( ) ][ ( )() ] ()( ) A ⎛⎞ + + ⎜⎟ ⎝⎠ + + 27. 100 01 011 1 1 001 ( )() =
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
SSM: Elementary Linear Algebra Section 2.1 47 29. 0000 1200 0238 0 0430 1238 () == 31. 12 7 3 01 4 1 112 3 6 00 2 7 00 0 3 ( )() 33. Expand along the third column. 22 0 0 1 1 sin( ) cos( ) sin( ) cos( ) cos( ) sin( ) cos( ) sin( ) sin( ) cos( ) sin( ) cos( ) sin ( ) ( cos ( )) sin ( ) cos ( ) θθ θθθθ −= −+ =− =+ = 35. 11 1 1 01 f M in both matrices, so expanding along the first row or column gives that 21 . dd λ True/False 2.1 (a) False; the determinant is ad bc . (b) False; 100 10 0101 001 (c) True; if i + j is even then 1 ij a + and 1 . ij ij ij CM M + = Similarly, if 1 , ij ij ij M + = then 11 + so i + j must be even. (d) True; let , abc A bd e cef ⎡⎤ ⎢⎥ = ⎣⎦ then 12 1 ( ) be Cb f c e cf and 21 1 ( ) . bc f c e ef Similar arguments show that 23 32 CC = and 13 31 . = (e) True; the value is det( A ) regardless of the row or column chosen.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/22/2011 for the course MAC AEPH114 taught by Professor Knutt during the Spring '11 term at McGill.

Page1 / 17

ch02 - Chapter 2 Determinants Section 2.1 Exercise Set 2.1...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online