Multiplying two matrices together is more complicated than addition, subtraction or scalar multiplication. When matrices are multiplied together each element must be multiplied by a specific series of elements in the other matrix then these series are added together to create a single element in the resulting matrix. Each element is made from the a slightly different series creating a matrix of results.

The pattern of multiplication and addition that this creates is very specific and creates a characteristic result. This makes matrix multiplication stand out from the multiplication of other familiar entities as scalars and vectors.

To multiply matrix A by matrix B each element must be treated differently according to a set pattern. To find the first element of the resulting matrix C, C00 each element in the first row of matrix A must be multiplied by an element in the first column of the matrix B in order. The total of each of these products is then found giving the value of the first element in the resulting matrix C.
For example:

• The first element in the first row of matrix A is multiplied by the first element in the first column of B (A00xB00).
• Then the second element in the first row of A is multiplied by the second element in the first column of B (A01xB10).
• The third and final element of the first row of A is multiplied by the third element in the first column of B (A02xB20).
• The sum of these products is calculated to give the first element of the of matrix C (C00 = (A00xB00)+(A01xB10)+(A02xB20)).

This pattern continues for the next element C01:

• The first element of the first row of A is multiplied by the first element of the second column of B (A00B01),
• The second element of the first row of A is multiplied by the second element of the second column of B,
• The third and last element in the first row of A is multiplied by the third element in the second column of B (A02xB21),
• The sum of the product is then calculated (C01 = (A00xB01)+(A01xB11)+(A02xB21)).

The pattern continues for C10 (C10 = (A10xB00)+(A11xB10)+(A12xB20)) and C11 (C11 = (A10xB01)+(A11xB11)+(A12xB21)).

Matrix A multiplied by Matrix B:



Numerical example:



It may not be obvious at first but on closer inspection you can see that these rules require the number of Columns of matrix A must equal the number of Rows in matrix B so that the resulting matrix is determinable. This means only certain matrices with the correct dimensions can be multiplied together if we want to find a result.

For example:


Because there is no element in matrix B to multiply with A02 or A12 the resulting sum for every element in C is always incomplete making the values and the resulting matrix itself nonsense and therefore indeterminate.

Assuming the basic condition that the number of columns is equal to the number of rows in the second matrix, matrices have been shown to exhibit many interesting properties. With classical numbers are multiplied by 1 the number stays the same. A similar property is also shown by matrices.

The Unit Matrix



When a matrix is multiplied by a matrix called the "unit matrix" it remains unchanged.



The zeros in each column cancel out the other terms in each of the elements of the resulting matrix. The unit matrix can be modified to change the resulting matrix in predictable ways as each non zero term in the unit matrix acts on a single specific column of elements in the matrix it is multiplied with. This can be very usefull if you wish to perform a transformation on the data in a matrix for example if you create a matrix with the x,y,z coordinates of a 3d point and you multiply it with this modified unit matrix:



The resulting matrix is identical accept for the fact that the x coordinate is now negative.


This has the effect of reflecting that point on the y axis. Modifying other values in the unit matrix can create reflection about other axis:

Matrices that perform a specific task on a set of coordinates are called matrix operators. There are a variety of matrix operators which are designed specifically to act on a set of coordinates reflection, translation, rotation and distortion. Each one uses the properties of matrix multiplication to function. Although operations such as these can be accomplished using basic algebraic techniques matrix manipulation is often used when dealing with large sets of coordinates as is simplifies and organizes the calculations involved.

Multiplying Matrices Using Actionscript

Matrix Multiplication Function Using A 1D Array In Actionscript (4 2d points)

first_matrix = new Array (20,150, 80,60, 95,130, 220,110);
second_matrix = new Array (2,5,0,3);

matrix_mult (my_point_matrix, y_trans_matrix);


fucntion = matrix_mult (matrixA:Array, martrixB:Array):Array {
var result_matrix:Array = new Array();

result_matrix[0] = matrixA[0]*martrixB[0] + matrixA[1]*martrixB[2];
result_matrix[1] = matrixA[0]*martrixB[1] + matrixA[1]*martrixB[3];

result_matrix[2] = matrixA[2]*martrixB[0] + matrixA[3]*martrixB[2];
result_matrix[3] = matrixA[2]*martrixB[1] + matrixA[3]*martrixB[3];

result_matrix[4] = matrixA[4]*martrixB[0] + matrixA[5]*martrixB[2];
result_matrix[5] = matrixA[4]*martrixB[1] + matrixA[5]*martrixB[3];

result_matrix[6] = matrixA[6]*martrixB[0] + matrixA[7]*martrixB[2];
result_matrix[7] = matrixA[6]*martrixB[1] + matrixA[7]*martrixB[3];

return result_matrix;
}

Matrix Multiplication Function Using A 2 Dimentional Array In Actionscript (4 2d points)

var first_matrix:Array = new Array ();
first_matrix [0][0] = 20;
first_matrix [0][1] = 150;

first_matrix [1][0] = 80;
first_matrix [1][1] = 60;

first_matrix [2][0] = 95;
first_matrix [2][1] = 130;

first_matrix [3][0] = 220;
first_matrix [3][1] = 110;

var second_matrix = new Array ();
second_matrix = [0][0]= 2;
second_matrix = [0][1]= 5;

second_matrix = [1][0]= 0;
second_matrix = [1][1]= 3;

matrix_Multiply(first_matrix:Array, second_matrix:Array)

function = matrix_Multiply(A:Array,B:Array) {    
var C = new Array(new Array(), new Array());    

C[0][0] = A[0][0]*B[0][0]+A[0][1]*B[1][0];
C[0][1] = A[0][0]*B[0][1]+A[0][1]*B[1][1];

C[1][0] = A[1][0]*B[0][0]+A[1][1]*B[1][0];
C[1][1] = A[1][0]*B[0][1]+A[1][1]*B[1][1];

C[2][0] = A[2][0]*B[0][0]+A[2][1]*B[1][0];
C[2][1] = A[2][0]*B[0][1]+A[2][1]*B[1][1];

return C;
}

Matrix Multiplication Function Using A 2D Array In Actionscript (4 3d points)

var first_matrix:Array = new Array ();
first_matrix [0][0] = 20;
first_matrix [0][1] = 150;
first_matrix [0][0] = 80;

first_matrix [1][0] = 60;
first_matrix [1][1] = 95;
first_matrix [1][2] = 130;

first_matrix [2][0] = 220;
first_matrix [2][1] = 110;
first_matrix [2][2] = 20;

first_matrix [3][0] = 280;
first_matrix [3][1] = 160;
first_matrix [3][2] = 90;

var second_matrix = new Array ();
second_matrix = [0][0]= 2;
second_matrix = [0][1]= 5;
second_matrix = [0][2]= 0;

second_matrix = [1][0]= 3;
second_matrix = [1][1]= 2;
second_matrix = [1][2]= 5;

second_matrix = [2][0]= 8;
second_matrix = [2][1]= 1;
second_matrix = [2][2]= 3;

matrix_Multiply(first_matrix:Array, second_matrix:Array)

function = matrix_Multiply(A:Array,B:Array) {    
var C = new Array(new Array(), new Array(), new Array());    

C[0][0] = A[0][0]*B[0][0]+A[0][1]*B[1][0]+A[0][2]*B[2][0];
C[0][1] = A[0][0]*B[0][1]+A[0][1]*B[1][1]+A[0][2]*B[2][1];
C[0][2] = A[0][0]*B[0][2]+A[0][1]*B[1][2]+A[0][2]*B[2][2];

C[1][0] = A[1][0]*B[0][0]+A[1][1]*B[1][0]+A[1][2]*B[2][0];
C[1][1] = A[1][0]*B[0][1]+A[1][1]*B[1][1]+A[1][2]*B[2][1];
C[1][2] = A[1][0]*B[0][2]+A[1][1]*B[1][2]+A[1][2]*B[2][2];

C[2][0] = A[2][0]*B[0][0]+A[2][1]*B[1][0]+A[2][2]*B[2][0];
C[2][1] = A[2][0]*B[0][1]+A[2][1]*B[1][1]+A[2][2]*B[2][1];
C[2][2] = A[2][0]*B[0][2]+A[2][1]*B[1][2]+A[2][2]*B[2][2];

C[3][0] = A[3][0]*B[0][0]+A[3][1]*B[1][0]+A[3][2]*B[2][0];
C[3][1] = A[3][0]*B[0][1]+A[3][1]*B[1][1]+A[3][2]*B[2][1];
C[2][2] = A[3][0]*B[0][2]+A[3][1]*B[1][2]+A[3][2]*B[2][2];

return C;
}

To learn more about matrices in flash see:

[Matrix Knowledgebase]
[Adding Matrices]
[Subtracting Matrices]
[Multiplying A Matrix By A Scalar]