maxima matrix (%i1 (%o1 (%i2 (%o2 matrix([1,2,3],[4,5,6],[7,8,9]; ( matrix([a,b,c,d],[e,f,g,h]; a b c d e f g h matrix [ ] ma

maxima Contents 1. 2 1.1. 2 1.2. 2 1.3. ( 2 1.4. 3 1.5. 3 1.6. 3 1.7. 4 1.8. 4 1.9. 5 1.10. 5 1.11. 5 1.12. 6 1.13. 6 2. 7 2.1. 7 2.2. 8 2.3. 9 2.4. 9 2.5. 10 3. 12 Date: 2005/10/5. 1

2 1. 1.1.. maxima matrix (%i1 (%o1 (%i2 (%o2 matrix([1,2,3],[4,5,6],[7,8,9]; 1 2 3 ( matrix([a,b,c,d],[e,f,g,h]; a b c d e f g h matrix [ ] 1 1.2.. matrix maxima maxima : (%i3 (%o3 (%i4 (%o4 A : matrix([1,2,3],[4,5,6],[7,8,9]; 1 2 3 ( B : matrix([a,b,c,d],[e,f,g,h]; a b c d e f g h A, B A B A B (%i5 A; (%o5 1 2 3 (%i6 ( B; a b c d (%o6 e f g h 1.3. (. A A A1 (%i7 A1 : A; (%o7 1 2 3 (%i8 A1; (%o8 1 2 3 A A1 A A1 A1 A

maxima 3 1.4.. A C ( A maxima copymatrix : (%i9 C :copymatrix(a; (%o9 1 2 3 (%i10 C; (%o10 1 2 3 A C A C 1.5.. maxima + A, B, A1, C A, B maxima kill (%i11 kill(a, B, A1, C; (%o11 DONE (%i12 ( A : matrix([1,2],[3,4]; 1 2 (%o12 3 4 (%i13 ( B : matrix([a,b],[c,d]; a b (%o13 c d (%i14 ( C : A + B; a + 1 b + 2 (%o14 c + 3 d + 4 A B 2 2- A + B C (%i15 ( A + B; a + 1 b + 2 (%o15 c + 3 d + 4 1.6... (%i16 ( A. B; 2c + a 2d + b (%o16 4c + 3a 4d + 3b (%i17 ( D : matrix([a,b,c],[d,e,f]; a b c (%o17 d e f

4 (%i18 E : matrix([p,q],[r,s],[t,u]; p q (%o18 r s t u (%i19 ( F : D. E; ct + br + ap cu + bs + aq (%o19 ft + er + dp fu + es + dq 2 2- A, B AB 2 3- D 3 2- E DE F (%i20 D. A; incompatible dimensions - cannot multiply -- an error. Quitting. To debug this try DEBUGMODE(TRUE; 1.7.. * ( kill(all (%i21 kill(all; (%o0 DONE (%i1 ( A : matrix([a,b,c],[d,e,f]; a b c (%o1 d e f (%i2 ( B : 3 * A; 3a 3b 3c (%o2 3d 3e 3f 1.8.. A A maxima ˆˆ (%i3 kill(all; (%o0 DONE (%i1 A : matrix([2,3,4],[-1,0,3],[2,1,-2]; (%o1 1 0 3 2 1 2 (%i2 A^^3; 26 36 48 (%o2 12 2 36 24 12 22 A 3 (%i3 A. A. A; 26 36 48 (%o3 12 2 36 24 12 22

maxima 5 ( 1.9.. maxima maxima transpose (%i4 A; (%o4 (%i5 (%o5 (%i6 (%o6 (%i7 (%o7 1 0 3 2 3 4 2 1 2 transpose(a; 2 1 2 3 0 1 4 3 2 ( C : matrix([a,b,c],[d,e,f]; a b c d e f transpose(c; a d b e c f 1.10.. A maxima echelon (%i8 kill(all; (%o0 DONE (%i1 A : matrix([1,2,3],[4,5,6],[7,8,9]; (%o1 1 2 3 (%i2 echelon(a; (%o2 1 2 3 0 1 2 0 0 0 (%i3 B : matrix([1,2,3],[4,5,6],[7,8,0]; (%o3 1 2 3 7 8 0 (%i4 C : echelon(b; (%o4 1 8 7 0 0 1 14 0 0 1 maxima 1.11.. A echelon(a A (rank maxima rank (%i5 rank(a; (%o5 2 (%i6 rank(b;

6 (%o6 3 A, B 1.12.. n A n A A 1 maxima invert (%i7 (%o7 invert(b; 16 8 9 9 1 9 14 9 7 2 9 9 1 2 9 9 1 9 invert (%i8 invert(a; Division by 0 -- an error. Quitting. To debug this try DEBUGMODE(TRUE; rank Aˆˆ(-1 A (%i9 (%o9 B^^(-1; 16 8 9 9 1 9 14 9 7 2 9 9 1 2 9 9 1 9 1.13.. A maxima determinant (%i10 determinant(a; (%o10 0 (%i11 determinant(b; (%o11 27 (%i12 determinant(b^^(-1; (%o12 1 27

maxima 7 2. 2.1.. i A[i] maxima row row(a,i (%i13 kill(all; (%o0 DONE (%i1 A : matrix([a,b,c],[d,e,f],[g,h,i]; (%o1 a b c d e f g h i (%i2 A[1]; (%o2 [a, b, c] (%i3 A[2]; (%o3 [d, e, f] (%i4 A[3]; (%o4 [g, h, i] (%i5 row(a,1; (%o5 ( a b c (%i6 row(a,2; (%o6 ( d e f (%i7 row(a,3; (%o7 ( g h i 2 A[i] row(a,i A[1] [a,b,c] a, b, c row(a,1 (a b c 1 3- A 2 2 (%i8 A[2] : 2 * A[2]; (%o8 [2d, 2e, 2f] (%i9 A; (%o9 a b c 2d 2e 2f g h i A 2 2 A A row A row(a,2 : (1/2 * row(a,2 (%i10 row(a,2 : (1/2 * row(a,2; Improper value assignment: ROW (A, 2 FALSEfalse -- an error. Quitting. To debug this try DEBUGMODE(TRUE; (%i11 A[2] : (1/2 * A[2]; (%o11 [d, e, f] (%i12 A;

8 (%o12 a b c d e f g h i A 1 2 3 (%i13 A[1] : A[1] + 3 * A[2]; (%o13 [3d + a, 3e + b, 3f + c] (%i14 A; 3d + a 3e + b 3f + c (%o14 d e f g h i (%i15 A[1] : A[1] - 3 * A[2]; (%o15 [a, b, c] (%i16 A; (%o16 a b c d e f g h i A 1 1 B (%i17 B : A[1]; (%o17 [a, b, c] (%i18 A[1] : A[3]; (%o18 [g, h, i] (%i19 A[3] : B; (%o19 [a, b, c] (%i20 A; (%o20 g h i d e f a b c 2.2.. A i transpose(transpose(a[i] col(a[i] (%i21 kill(all; (%o0 DONE (%i1 A : matrix([r,s,t],[u,v,w],[x,y,z]; (%o1 r s t u v w x y z (%i2 transpose(transpose(a[1]; (%o2 (%i3 (%o3 r u x col(a,1; r u x

maxima 9 transpose(transpose(a[1] col(a,1 3 1- transpose 2.3.. A (i,j- A[i][j] A[i,j] (%i4 kill(all; (%o0 DONE (%i1 A : matrix([1,2,3,4],[5,6,7,8],[9,10,11,12]; (%o1 1 2 3 4 5 6 7 8 9 10 11 12 (%i2 A[2][3]; (%o2 7 (%i3 A[2,3]; (%o3 7 A (2,3 A (2,3 x? A[2][3] : x OK (%i4 A[2,3] : x; (%o4 x (%i5 A; (%o5 1 2 3 4 5 6 x 8 9 10 11 12 A (3,2- y A[3,2] : y (%i6 A[3,2] : y; (%o6 y (%i7 A; (%o7 1 2 3 4 5 6 x 8 9 y 11 12 2.4.. maxima x + 2y + 3z = 4, 5x + 6y + 7z = 8, 9x + 10y + 11z = 0 maxima eq1 (%i8 kill(all; (%o0 DONE (%i1 eq1 : [x+2*y+3*z=4, 5*x+6*y+7*z=8, 9*x+10*y+11*z=12]; (%o1 [3z + 2y + x = 4, 7z + 6y + 5x = 8, 11z + 10y + 9x = 12] eq1 (x,y,z A maxima coefmatrix

10 (%i2 xx : [x,y,z]; (%o2 [x, y, z] (%i3 A : coefmatrix(eq1, xx; (%o3 1 2 3 5 6 7 9 10 11 xx [x,y,z] (%i3 A : coefmatrix([x+2*y+3*z=4, 5*x+6*y+7*z=8, 9*x+10*y+11*z=12],[x,y,x]; Ax = b (A b maxima augcoefmatrix (A b (A b (%i4 (%o4 Ab : augcoefmatrix(eq1, xx; 1 2 3 4 5 6 7 8 9 10 11 12 rank(a rank(ab maxima (%i5 echelon(ab; (%o5 1 2 3 4 0 1 2 3 0 0 0 0 (%i6 rank(a; (%o6 2 (%i7 rank(ab; (%o7 2 rank(a = rank(ab = 2 eq1 n = 3 rank(a = 2 3-2 = 1 2.5.. maxima eq1 maxima solve (%i8 solve(eq1, xx; Dependent equations eliminated: (3 (%o8 [[x = %R1 2, y = 3 2%R1, z = %R1]] %R1 solve (%i9 kill(all; (%o0 DONE

maxima 11 (%i1 eq2 : [x + 2*y = 3, 4*x + 5*y = 6, 7*x + 8*y = 0]; (%o1 [2y + x = 3, 5y + 4x = 6, 8y + 7x = 0] (%i2 xx : [x,y]; (%o2 [x, y] (%i3 solve(eq2, xx; Inconsistent equations: (1 -- an error. Quitting. To debug this try DEBUGMODE(TRUE; (%i4 A : coefmatrix(eq2, xx; (%o4 1 2 4 5 7 8 (%i5 Ab : augcoefmatrix(eq2, xx; (%o5 1 2 3 4 5 6 7 8 0 (%i6 echelon(ab; (%o6 1 8 7 0 0 1 14 0 0 1 (%i7 rank(a; (%o7 2 (%i8 rank(ab; (%o8 3 rank(a rank(ab eq2

12 3. maxima, maxima maxima

E-mail address: kenrou@yo.rim.or.jp maxima 13