MAPLE: Matrix Algebra

1. Whenever doing linear algebra on maple, first enter

   > with(linalg):

2. For a list of commands available in the linalg package

   > ?linalg

3. For help on any of the linalg commands, enter ?command, for example:

   > ?det

   > ?mulrow

   > ?addrow

   > ?matrix

   > ?vector

   > ?multiply

   > ?augment

4. To define a matrix A, say 6 by 4 (6 rows and 4 columns) without specifying its entries:

   > A:= matrix(6,4);

5. Example: to create matrix A while specifying its entries:

   A =	6	3	-2	1
   	8	0	7	2
   	9	5	4	-3

   > A:= matrix([[6,3,-2,1],[8,0,7,2],[9,5,4,-3]]);

   Spaces are allowed but not required. Also, the command may span several lines.

   > A[3,4]; shows that -3 is stored in row 3 column 4 of A

   > A[3,4]:=9; stores 9 in row 3 column 4 of A

6. Some common linear algebra commands. Suppose we have matrices A, B, and C:

   A = 6 3 -2 1 8 0 7 2 9 5 4 -3

   B = 7 2 0 -5 3 4 2 1 9 6 -7 2

   C = 7 2 0 -5 3 4 2 1 9

   > A:= matrix([[6,3,-2,1],[8,0,7,2],[9,5,4,-3]]);

   > B:= matrix([[7,2,0],[-5,3,4],[2,1,9],[6,-7,2]]);

   > C:= matrix([[7,2,0],[-5,3,4],[2,1,9]]);

   > det(C); gives the determinant of square matrix C

   > inverse(A); gives the inverse of matrix A

   > rank(A); gives the rank of matrix A

   > gaussjord(A); gives the Gauss-Jordan canonical form of matrix A

   > augment(A,C); augments two matrices A and C

   > multiply(A,B); matrix product AB

   > multiply(B,A); matrix product BA

7. To view an existing matrix:

   > evalm(A);

8. To multiply row r of matrix A by k:

   > A:=mulrow(A,r,k);

9. To multiply column c of matrix A by k:

   > A:=mulcol(A,c,k);

10. To replace row r₁ of matrix A by k × (row r₁) + (row r₂), where k is a constant:

    > A:=addrow(A,r₁,r₂,k);

11. To obtain the characteristic polynomial of a square matrix C:

    > p:=charpoly(C,x); calls the resulting polynomial p

12. To obtain eigenvalues of a square matrix C:

    > eigenvals(C,'radical');

13. To obtain eigenvectors and eigenvalues of a square matrix C:

    > eigenvects(C,'radical');

14. TO SOLVE A LINEAR SYSTEM of n equations for its n variables, you may define an n x (n+1) augmented matrix and then use the command gaussjord. For example, suppose we wish to solve the following system of 3 linear equations for its 3 unknowns x, y, and z:
    6x + 3y - 2z = 1

    8x + 7z = 2

    9x + 5y + 4z = -3

    > M:= matrix([[6,3,-2,1],[8,0,7,2],[9,5,4,-3]]); gives

    M =	6	3	-2	1
    	8	0	7	2
    	9	5	4	-3

    > gaussjord(M);

    1	0	0	142 / 197
    0	1	0	-289 / 197
    0	0	1	-106 / 197

    As long as the identity matrix appears in the coefficient matrix portion, then the rightmost column gives the solution x, y, z.

    If you use a decimal point anywhere, the solution will be given in decimal form.

    Alternatively, you can solve the linear system Ax = b where A is an n x n coefficient matrix and b is the n-term constant vector by:

    > A:= matrix([[6,3,-2],[8,0,7],[9,5,4]]); defines coefficient matrix A

    A =	6	3	-2
    	8	0	7
    	9	5	4

    > b:= vector([1,2,-3]); defines constant vector b

    > linsolve(A,b); gives the solution of system Ax = b in vector form

    > det(A); gives the determinant of coefficient matrix A

    > AI:=inverse(A); gives the inverse A^-1 of coefficient matrix A and calls it AI

    > multiply(AI,b); multiplies A^-1b, which is the solution x of Ax = b

15. The linalg package has many other commands. For a complete list, enter

    > ?linalg;

Written and Maintained by

Prof. Kevin G. TeBeest
Applied Mathematics
Kettering University

Last modified: 07/23/07

Copyright © 1997-2008 Kevin G. TeBeest. All rights reserved.