Matrices
From GeoGebra Manual
GeoGebra supports real matrices, which are represented as a list of lists that contain the rows of the matrix.
Example: In GeoGebra, {{1, 2, 3}, {4, 5, 6}, {7, 8, 9}} represents the 3x3 matrix
To display a matrix using LaTeX formatting in the Graphics View, use the FormulaText command or drag and drop the matrix definition from Algebra View to Graphics View.
Example: In the Input Bar type
FormulaText[{{1, 2, 3}, {4, 5, 6}, {7, 8, 9}}]
to display the matrix using LaTeX formatting.
Accessing Elements of Matrices
To access particular elements of a matrix you can use the Element Command or the simplified syntax shown in the example below:
Example: Let
matrix={{1, 2}, {3, 4}}
, then: matrix(1, 1)
returns the first element in the first line: 1matrix(2, 2)
,matrix(1,2)
,matrix(2,1)
andmatrix(1,1)
all return the second element of the second line: 4. In general,
matrix(i, j)
, where i and j are integers, returns the element of the matrix that occupies the ith row and the jth column.
Matrix Operations
Matrix operations are operations with lists, so the following syntaxes produce the described results.
Note: Some syntaxes can represent operations which are not defined in the same way in the matrices set.
Addition and subtraction

Matrix1 + Matrix2
: adds the corresponding elements of two compatible matrices. 
Matrix1 – Matrix2
: subtracts the corresponding elements of two compatible matrices.
Multiplication and division

Matrix * Number
: multiplies each element of Matrix by the given Number. 
Matrix1 * Matrix2
: uses matrix multiplication to calculate the resulting matrix.
 Note: The rows of the first and columns of the second matrix need to have the same number of elements.
 Example: {{1, 2}, {3, 4}, {5, 6}} * {{1, 2, 3}, {4, 5, 6}} yields the matrix {{9, 12, 15}, {19, 26, 33}, {29, 40, 51}}.

2x2 Matrix * Point (or Vector)
: multiplies the Matrix by the given Point / Vector and yields a point.
 Example: {{1, 2}, {3, 4}} * (3, 4) yields the point A = (11, 25).

3x3 Matrix * Point (or Vector)
: multiplies the Matrix by the given Point / Vector and yields a point.
 Example: {{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * (1, 2) gives you the point A = (8, 20).
 Note: This is a special case for affine transformations where homogeneous coordinates are used: (x, y, 1) for a point and (x, y, 0) for a vector. This example is therefore equivalent to:
{{1, 2, 3}, {4, 5, 6}, {0, 0, 1}} * {1, 2, 1}
.

Matrix1 / Matrix2
: Divides each element of Matrix1 by the corresponding element in Matrix2.
 Note: However, GeoGebra supports the syntax
Matrix1 * Matrix2 ^(1)
.
Other operations
The section Matrix Commands contains the list of all available commands related to matrices, such as:
 Determinant[Matrix]: calculates the determinant for the given matrix.
 Invert[Matrix]: inverts the given matrix
 Transpose[Matrix]: transposes the given matrix
 ApplyMatrix[Matrix,Object]: apply affine transform given by matrix on object.
 ReducedRowEchelonForm[Matrix]: converts the matrix to a reduced rowechelon form
Comments
Note: See the official forum for a more detailed discussion about the multiplication of matrices.