Question 11

Explanation

Scale Transformation Matrix

Scale matrix $(T_1)$ with factor $2$ on the x-axis and $0.5$ on the y-axis

Shear Transformation Matrix

Shear matrix along the x-axis with factor $1$ $(T_2)$

Composition Matrix

The composition matrix $(T_2 \circ T_1)$ is $M_2 \times M_1$

Calculating Initial Area

A square with side $2$ units has an initial area of

Determining Area Change Scale Factor

The determinant of the composition matrix $M_2 \cdot M_1$ gives the area change scale factor

Area After Transformation

The area after transformation is calculated by

Although scale and shear transformations occurred, the area of the square remains the same because the determinant of the composition matrix is $1$

Therefore, the area of the square after both transformations is $4$ square units