Consider the matrix = \(\left[ {\begin{array}{*{20}{c}} 2&0&2\\ 0&1&1\\ 0&0&1 \end{array}} \right]\) representing a set of planar (2D) geometric transformations in homogeneous coordinates. Which of the following statements about the matrix M is True ?

Concept:

Scaling:

In this, the size of an object is increased or decreased by changing the dimension. Representation of the scaling matrix is:

 \(\left[ {\begin{array}{*{20}{c}} a&0&0\\ 0&b&0\\ 0&0&1 \end{array}} \right]\)

Translation:

It moves an object from its original position to a new position by adding a translation factor with the original coordinates. It does not change the size. Translation matrix is represented as :

\(\left[ {\begin{array}{*{20}{c}} 1&0&a\\ 0&1&b\\ 0&0&1 \end{array}} \right]\)

Explanation:

Given scalar matrix

 \(\left[ {\begin{array}{*{20}{c}} 2&0&0\\ 0&1&0\\ 0&0&1 \end{array}} \right]\)

Translation matrix will be:

\(\left[ {\begin{array}{*{20}{c}} 1&0&1\\ 0&1&1\\ 0&0&1 \end{array}} \right]\)

If we do, first translation of (1,1), then scaling of a vector (2,1), then we get the original matrix. So, multiply scalar matrix with translation matrix of the specified vector and finally, we get the matrix M.