soln2 - MATH 235/W08 Determinants Solutions to Assignment 2...

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Unformatted text preview: MATH 235/W08 Determinants Solutions to Assignment 2 Hand in questions 1,2,3,4,5,6,7,8 Several solutions are found using MATLAB. The MATLAB file http://orion.math.uwaterloo.ca/hwolkowi/henr contains a diary command and a echo on command. 1. Evaluate the determinants of the following matrices: (a) 1 i 1 + i 2- i 3- 1 + i 3 1- i (b) 2 3 1- 1 . 56- 1 . 58 4 . 34 4 9 102 (c) a b c 2 a + 3 d 2 b + 3 e 2 c + 3 f d e f , where a, b, c, d, e, f C . (d) 6- 1 9 3 2 3 7 8 3 2 9 4 5 5 1 BEGIN SOLUTION: disp(solutions for assign 2 in W08) solutions for assign 2 in W08 disp( ) disp(Prob 1. Evaluate the det of the following four matrices) 1 Prob 1. Evaluate the det of the following four matrices A1=[ 1 i 1+i 2-i 3-1+ i 3 1-i A1 = 1.0000 0 + 1.0000i 1.0000 + 1.0000i 2.0000 - 1.0000i 3.0000-1.0000 + 1.0000i 3.0000 1.0000 - 1.0000i ] det(A1) ans = 15.0000 - 1.0000i disp(or by expanding along row 2 we get ) or by expanding along row 2 we get-(2-i)*det([i 1+i;3 1-i]) +3*det([1 1+i;-1+i 1-i]) ans = 15.0000 - 1.0000i A2=[ 2 3 pi 1-1.56-1.58 4.34 4 9 102 A2 = 2.0000 3.0000 3.1416 1.0000-1.5600-1.5800 4.3400 4.0000 9.0000 102.0000 ] det(A2) ans = 2 72 disp(exchange rows 1,2 and rows 3,4 to get a triangular matrix) exchange rows 1,2 and rows 3,4 to get a triangular matrix A2copy=A2([2 1 4 3],:) A2copy = 1.0000-1.5600-1.5800 4.3400 2.0000 3.0000 3.1416 9.0000 102.0000 4.0000 prod(diag(A2copy)) ans = 72 pause a=sym(a) a = a b=sym(b) b = b c=sym(c) c = c d=sym(d) d = 3 d e=sym(e) e = e f=sym(f) f = f A3=[ a b c 2*a+3*d 2*b+3*e 2*c+3*f d e f A3 = [ a, b, c] [ 2*a+3*d, 2*b+3*e, 2*c+3*f] [ d, e, f] ] disp(row 2 is a linear combination of rows 1 and 3 - so det is 0 - verify) row 2 is a linear combination of rows 1 and 3 - so det is 0 - verify det(A3) ans = A4=[ 6-1 A4 = 6-1 4 9 3 2 3 7 8 3 2 9 4 5 5 1 9 3 2 3 7 8 3 2 9 4 5 5 1 ] det(A4) ans =-144 disp(exchange two rows) exchange two rows A4copy=A4([4 1 2 3 5],:) A4copy = 4 6-1 9 3 2 3 7 8 3 2 9 5 5 1 disp(exchange two cols) exchange two cols disp(the sign after the two exchanges is +) the sign after the two exchanges is + A4copy=A4copy(:,[3 1 2 4 5]) A4copy = 4-1 6 2 9 3 3 7 3 8 2 9 5 5 1 det(A4copy) ans = 5 144 disp(by expanding along row 1 repeatedly: det is product along diagonal ) by expanding along row 1 repeatedly: det is product along diagonal prod(diag(A4copy)) ans = 144 END SOLUTION. 2. Find the value of the determinants of the following matrices. Explain your answer in terms of cofactor expansion: (a) 1- 1 1- 1 (b) The 100 100 matrix A below: A = 1 1 ....
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This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.

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soln2 - MATH 235/W08 Determinants Solutions to Assignment 2...

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