Matrix transformations are performed through matrix multiplication of a point matrix by a transformation matrix.
The transformation matrix affects the point matrix, creating a new point matrix.
[T][P]=[N] where:
 [T] is the transformation matrix
 [P] is the point matrix being transformed
 [N] is the new transformed point matrix
The matrices are multiplied in this order in order to obtain a new point matrix and not a larger matrix.
A point matrix, P, takes the form of an n x 1 though generally, 2 x 1 and 3 x 1 matrices are
generally used as they model 2D and 3D systems respectivey. The value P_{1,1} is the x coordinate and the
value P_{2,1} is the y coordinate. In a 3D point matrix the value of P_{3,1} is the z coordinate.
For example:
 is the form 
 Therefore, the coordinates of this point matrix
are: (4,3,1) 
A transformation matrix is a square matrix which has an affect on a point matrix so as to transform it's position
on it's coordinate axis when they are multiplied. The no. of rows of the point matrix determines
the dimensions of the transformation matrix as the no. of columns of the transformation matrix must equal the no.
of rows of the point matrix in order to allow matrix multiplication to occur as previously discussed in
Basic Matrix Arithmetic.
For example:  
This transformation matrix will affect a reflection in the line y=x for the point matrix, as shown in the example below: 
Transformation Matrix 
Point Matrix   Multiplication   New Point Matrix 
  = 
 =  

The transformation matrix has caused the point matrix to reflect in the line y=x creating the coordinate
(3,4) from the coordinate (4,3). 
A list of some 2D transformation matrices and their affects can be found in the section
2D Matrix Rotations and Reflections.
The point matrix
denoting the coordinates (4,3) will be used as an example throughout this page in order to emphasise the changes
that are taking placing in the point matrices under transformation matrices.
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Matrix rotation and reflection is achieved through matrix multiplication with specific matrices
which achieve certain rotations or reflections. This is effectively plotting the coordinates of the
previous matrix and then matrix multiplying it and ending up with new coordinates of the matrix at
another point on the graph.
There are a few rules which need to be followed when rotating or reflecting a matrix:
 The transformation matrix is always the leading matrix in the multiplication [T][P]=[N] where:
 [T] is the transformation matrix
 [P] is the point matrix
 [N] is the new matrix
 A point matrix is a n x 1 vector as it describes a point on a graph
 The no. of rows of the transformation matrix must equal the no. of columns of the point matrix.
The matrices listed below all perform different rotations/reflections:
Transformation Matrix  Affect of Transformation Matrix on Point Matrix  Example 
 This transformation matrix is the identity matrix.
When multiplying by this matrix, the point matrix is unaffected and the new matrix is exactly the same as the point matrix. 

 This transformation matrix creates a reflection in the xaxis.
When multiplying by this matrix, the x coordinate remains unchanged, but the y coordinate changes sign. 

 This transformation matrix creates a reflection in the yaxis.
When multiplying by this matrix, the y coordiante remains unchanged, but the x coordinate changes sign. 

 This transformation matrix creates a rotation of 180 degrees.
When multiplying by this matrix, the point matrix is rotated 180 degrees around (0,0). This changes the sign of both the x and y coordinates. 

 This transformation matrix creates a reflection in the line y=x.
When multiplying by this matrix, the x coordinate becomes the y coordinate and the yordinate becomes the x coordinate. 

 This transformation matrix rotates the point matrix 90 degrees clockwise.
When multiplying by this matrix, the point matrix is rotated 90 degrees clockwise around (0,0). 

 This transformation matrix rotates the point matrix 90 degrees anticlockwise.
When multiplying by this matrix, the point matrix is rotated 90 degrees anticlockwise around (0,0). 

 This transformation matrix creates a reflection in the line y=x.
When multiplying by this matrix, the point matrix is reflected in the line y=x changing the signs of both coordiantes and swapping their values. 

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The enlargement of a matrix can be achieved through matrix multiplication of a transformation matrix followed by
a point matrix: [T][P]=[N]
The section 2D Reflections and Rotations
shows how transforming with the identity matrix, doesn't appear to alter the coordiantes of the point matrix.
This is due to:
 the new x coordinate being constructed of the old x coordinate multiplied by one added to the old
y coordiante multiplied by zero
 the new y coordinate being constructed of the old y coordinate multiplied by one added to the old
x coordiante multiplied by zero
eg.
However, if the one's in this transformation matrix, were not ones and were another integer number, then the
value of the new point matrix would be an enlargement of that integer number, on the original matrix. This is
because the new x coordiante is the old one multiplied by T_{1,1} and the new ycoordinateis the old
one multiplied by T_{2,2}.
This means that the identity matrix really performs an enlargment of scale factor 1.
Below are some examples of transformation matrices which enlarge point matrices:
Transformation Matrix  Affect of Transformation Matrix on Point Matrix  Example 
 This transformation matrix is the identity matrix
multiplied by the scalar 6. When multiplying by this matrix, the point matrix is enlarged by a factor of 6 in the x and y directions. 

 This transformation matrix is the identity matrix but T_{1,1}
has been larged by a factor of 7 and T_{2,2} has been enlarged by a factor of 0. When multiplying by this matrix, the x coordinate is enlarged
by a factor of 7, whilst the y coordinate is enlarged by a factor of 0. 

 This transformation matrix is the identity matrix but T_{1,1}
has been larged by a factor of a and T_{2,2} has been enlarged by a factor of b. When multiplying by this matrix, the x coordinate is enlarged
by a factor of a, whilst the y coordinate is enlarged by a factor of b. 

 This transformation matrix creates a rotation and an enlargement.
When multiplying by this matrix, the point matrix is rotated 90 degrees anticlockwise around (0,0), whilst the x xoordinate of the new point matrix
is enlarged by a factor of 5 and the y coordinate of the new point matrix is enlarged by a factor of 7. 

The table above shows that it is the position and sign of elements in a rotation or reflection transformation matrix that decides the rotation or reflection
of the point matrix and not the value of the elements as enlarging the value of an element in a rotation matrix only enlarges the rotation. Enlarging a point
matrix by a factor in either the x or y direction requires that the nonzero elements of the transformation matrix be enlarged by that factor.
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