% Problem 1 format rat; % (a) B = [ 1 -2 0 0 -9 -8; 3 -5 -4 1 -29 -27; -1 2 0 1 11 11; -4 6 82 -6 32 26 ]; % (b) B(2,:) = B(2,:) - 3 * B(1,:) B(3,:) = B(3,:) + B(1,:) B(4,:) = B(4,:) + 4 * B(1,:) B(4,:) = B(4,:) + 2 * B(2,:) B([3 4],:) = B([4 3],:) % (c) B(3,:) = B(3,:) / 74 B(3,:) = B(3,:) + 2/37 * B(4,:) B(2,:) = B(2,:) - B(4,:) B(2,:) = B(2,:) + 4 * B(3,:) B(1,:) = B(1,:) + 2 * B(2,:) % (d) rref(B) % (e) % The solution of this system in parametric vector form is % x1 = 17 * x5 - 20 % x2 = 4 * x5 - 6 % x3 = 0 % x4 = -2 * x5 + 3 % x5 is free clear; % Problem 2 format rat % (a) syms a syms b % (b) A = [ 1 3; 3 8 ] rref(A) % (c) B = [ 8 -5 a; 4 1 b ] rref(B) % w1 = a/28 + 5*b/28 % w2 = 2*b/7 - a/7 clear; % Problem 3 format rat; % (a) A = [ 1 1 1 2 -2; -4 -3 -7 -17 4; 2 3 0 -3 -12; 3 2 2 7 15 ] rref(A) % (b) % The fourth column is not a pivot column % (c) % Suppose these vectors are v1, v2, v3, v4, v5. If we let the free variable x4 = 1, then a linear dependence could be -3 * v1 + 3 * v2 - 2 * v3 + v4 = 0 % (d) % The linear system Ax = 0 has nontrivial solution % (e) % Yes, since there is a pivot in every row clear; % Problem 4 format short % (a) A = [ cos(pi/10) -sin(pi/10); sin(pi/10) cos(pi/10) ] % (b) B = [ cos(pi/12) -sin(pi/12); sin(pi/12) cos(pi/12) ] A * B - B * A % (c) % rotation operations commute % (d) C = A * B t = acos(C(1,1)) format rat; t / pi % (e) format short; A1 = [ cos(-pi/10) -sin(-pi/10); sin(-pi/10) cos(-pi/10) ] A1 - inv(A) % (f) L0 = [ 1 0; 0 -1 ] L1 = A * L0 * A1 % (g) L1 * L0 - L0 * L1 % (h) D = L1 * L0 t = acos(D(1,1)) format rat; t / pi clear; % Problem 5 format rat; % (a) A = [ 2 1 8; 6 4 3; 5 2 5 ] B = [A eye(3,3)] C = rref(B) C(:,[4,5,6]) % (b) inv(A) clear; % Problem 6 format rat; % (a) A = [ 1 0 0 0; -10 5 0 0; 11 0 -3 0; -4 12 -7 2 ] B = [ -1 2 0 1; 2 2 3 3; 0 1 -1 -1; 2 -1 3 1 ] det(A) det(B) % (b) % Cofactor expansion % (c) C = A * B det(C) % (d) % det(C) = det(A) * det(B) clear; % Problem 7 % (a) syms a b c d A = [ a b; c d ] % (b) inv(A) % (c) syms e f g h i B = [ a b c d e f g h i ] inv(B) % (d) det(B) * inv(B)
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 1 -2
-1 2 0 1 11
-4 6 82 -6 32
Column 6
-8
-3
11
26
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 1 -2
0 0 0 1 2
-4 6 82 -6 32
Column 6
-8
-3
3
26
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 1 -2
0 0 0 1 2
0 -2 82 -6 -4
Column 6
-8
-3
3
-6
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 1 -2
0 0 0 1 2
0 0 74 -4 -8
Column 6
-8
-3
3
-12
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 1 -2
0 0 74 -4 -8
0 0 0 1 2
Column 6
-8
-3
-12
3
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 1 -2
0 0 1 -2/37 -4/37
0 0 0 1 2
Column 6
-8
-3
-6/37
3
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 1 -2
0 0 1 0 0
0 0 0 1 2
Column 6
-8
-3
0
3
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 -4 0 -4
0 0 1 0 0
0 0 0 1 2
Column 6
-8
-6
0
3
B =
Columns 1 through 5
1 -2 0 0 -9
0 1 0 0 -4
0 0 1 0 0
0 0 0 1 2
Column 6
-8
-6
0
3
B =
Columns 1 through 5
1 0 0 0 -17
0 1 0 0 -4
0 0 1 0 0
0 0 0 1 2
Column 6
-20
-6
0
3
ans =
Columns 1 through 5
1 0 0 0 -17
0 1 0 0 -4
0 0 1 0 0
0 0 0 1 2
Column 6
-20
-6
0
3
A =
1 3
3 8
ans =
1 0
0 1
B =
[8, -5, a]
[4, 1, b]
ans =
[1, 0, a/28 + (5*b)/28]
[0, 1, (2*b)/7 - a/7]
A =
1 1 1 2 -2
-4 -3 -7 -17 4
2 3 0 -3 -12
3 2 2 7 15
ans =
1 0 0 3 0
0 1 0 -3 0
0 0 1 2 0
0 0 0 0 1
A =
0.9511 -0.3090
0.3090 0.9511
B =
0.9659 -0.2588
0.2588 0.9659
ans =
0 0
0 0
C =
0.8387 -0.5446
0.5446 0.8387
t =
0.5760
ans =
11/60
A1 =
0.9511 0.3090
-0.3090 0.9511
ans =
1.0e-15 *
0 0
0 -0.1110
L0 =
1 0
0 -1
L1 =
0.8090 0.5878
0.5878 -0.8090
ans =
0 -1.1756
1.1756 0
D =
0.8090 -0.5878
0.5878 0.8090
t =
0.6283
ans =
1/5
A =
2 1 8
6 4 3
5 2 5
B =
Columns 1 through 5
2 1 8 1 0
6 4 3 0 1
5 2 5 0 0
Column 6
0
0
1
C =
Columns 1 through 5
1 0 0 -14/51 -11/51
0 1 0 5/17 10/17
0 0 1 8/51 -1/51
Column 6
29/51
-14/17
-2/51
ans =
-14/51 -11/51 29/51
5/17 10/17 -14/17
8/51 -1/51 -2/51
ans =
-14/51 -11/51 29/51
5/17 10/17 -14/17
8/51 -1/51 -2/51
A =
1 0 0 0
-10 5 0 0
11 0 -3 0
-4 12 -7 2
B =
-1 2 0 1
2 2 3 3
0 1 -1 -1
2 -1 3 1
ans =
-30
ans =
-12
C =
-1 2 0 1
20 -10 15 5
-11 19 3 14
32 7 49 41
ans =
360
A =
[a, b]
[c, d]
ans =
[ d/(a*d - b*c), -b/(a*d - b*c)]
[-c/(a*d - b*c), a/(a*d - b*c)]
B =
[a, b, c]
[d, e, f]
[g, h, i]
ans =
[ (e*i - f*h)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g), -(b*i - c*h)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g), (b*f - c*e)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g)]
[-(d*i - f*g)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g), (a*i - c*g)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g), -(a*f - c*d)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g)]
[ (d*h - e*g)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g), -(a*h - b*g)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g), (a*e - b*d)/(a*e*i - a*f*h - b*d*i + b*f*g + c*d*h - c*e*g)]
ans =
[e*i - f*h, c*h - b*i, b*f - c*e]
[f*g - d*i, a*i - c*g, c*d - a*f]
[d*h - e*g, b*g - a*h, a*e - b*d]