MAPLE HELPSHEETS: Matrix Algebra

MAPLE: Matrix Algebra

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Whenever doing linear algebra on maple, first enter
> with(linalg):
For a list of commands available in the linalg package
> ?linalg
For help on any of the linalg commands, enter ?command, for example:
> ?det
> ?mulrow
> ?addrow
> ?matrix
> ?vector
> ?multiply
> ?augment
To define a matrix A, say 6 by 4 (6 rows and 4 columns) without specifying its entries:
> A := matrix(6,4);
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

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

To view an existing matrix:
> evalm(A);
To multiply row r of matrix A by k:
> A := mulrow( A, r, k ) ;
To multiply column c of matrix A by k:
> A := mulcol( A, c, k ) ;
To replace row r₁ of matrix A by k × (row r₁) + (row r₂), where k is a constant:
> A := addrow( A, r₁, r₂, k ) ;
To obtain the characteristic polynomial of a square matrix C:
> p := charpoly(C, x) ; calls the resulting polynomial p
To obtain eigenvalues of a square matrix C:
> eigenvals(C,'radical');
To obtain eigenvectors and eigenvalues of a square matrix C:
> eigenvects(C,'radical');
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
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: 06/07/2012

Maple^® is a registered trademark of Waterloo Maple Software.

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