Matrix Transformation of a Geometric Shape

In this problem, we are given a transformation matrix that maps a unit square into a specific parallelogram. We need to find the values of lowercase a, b, c, and d.

Let's identify our matrices first. We have the transformation matrix T, and the resulting parallelogram matrix P.

A unit square in coordinate geometry is typically represented by its four vertices. We start at the origin zero zero, then go to one zero, then one one, and finally zero one.

The relationship is that the transformation matrix T multiplied by the unit square matrix U equals the parallelogram matrix P. Let's write that out.

Let's perform the matrix multiplication. We multiply the two-by-two matrix T by the two-by-four matrix U.

For the first column, we have zero times a plus one plus zero times two, which correctly gives zero in the resulting matrix. Now let's calculate the second column.

Comparing this to the second column of our result matrix, which is four and one, we can set up two equations.