A \({\bf{2}} \times {\bf{200}}\) data matrix \(D\) contains the coordinate of 200 points. Compute the number of multiplications required to transform these points using two arbitrary \({\bf{2}} \times {\bf{2}}\) matrices \(A\) and \(B\). Consider the two possibilities \[A\left( {BD} \right)\] and \(\left( {AB} \right)D\). Dicuss the implications of your results for computer graphics calculations.

For producing each entry in \(BD\), two multiplications are necessary. Since \(BD\) is a \(2 \times 200\) matrix, it will take \(2 \times 2 \times 200 = 800\;{\rm{multiplications}}\) to compute it. By the same reasoning, it will take \(2 \times 2 \times 200 = 800\;{\rm{multiplications}}\) to compute \(A\left( {BD} \right)\). Thus, to compute \(A\left( {BD} \right)\) from the beginning, it will take \(1600\;{\rm{multiplications}}\).

To compute the \(2 \times 2\) matrix \(AB\), it will take \(2 \times 2 \times 2 = 8\;{\rm{multiplications}}\), and to compute \(\left( {AB} \right)D\), it will take \(2 \times 2 \times 200 = 800\)multiplications. Thus, to compute \(\left( {AB} \right)D\) from the beginning, it will take \(8 + 800 = 808\) multiplications.

For computer graphics, calculations require the application of multiple transformations to data matrices. It is thus more efficient to compute the product of the transformation matrices before applying the result to the data matrix.

So, \(A\left( {BD} \right)\) requires 1600 multiplications, and \(\left( {AB} \right)D\) requires 808 multiplications. The first method uses twice as many applications.