Determine the single matrix M that describes an enlargement by factor 2 followed by a shear, factor 3 in the x-axis direction with y-axis invariant.

An enlargement by a factor of $k$ is represented by the matrix:

$$E = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$$

For an enlargement by factor 2, the matrix is:

$$E = \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$$

A shear with factor $k$ in the x-axis direction with the y-axis invariant is represented by the matrix:

$$S = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}$$

For a shear factor of 3 in the x-axis direction, the matrix is:

$$S = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix}$$

Since the enlargement is followed by the shear, the combined matrix M is found by multiplying the shear matrix by the enlargement matrix (in that order):

$$M = S \times E$$ $$M = \begin{bmatrix} 1 & 3 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$$

$$M = \begin{bmatrix} (1)(2) + (3)(0) & (1)(0) + (3)(2) \\ (0)(2) + (1)(0) & (0)(0) + (1)(2) \end{bmatrix}$$ $$M = \begin{bmatrix} 2 + 0 & 0 + 6 \\ 0 + 0 & 0 + 2 \end{bmatrix}$$ $$M = \begin{bmatrix} 2 & 6 \\ 0 & 2 \end{bmatrix}$$

The single matrix M is:

$$\begin{bmatrix} 2 & 6 \\ 0 & 2 \end{bmatrix}$$