Question Type 3: Finding the area of an image using matrix transformations Exercises

Write down the matrix representing an enlargement (scaling) with scale factor 2 about the origin.

Write down the matrix for a rotation of $90^\circ$ counterclockwise about the origin.

Write down the matrix that represents a horizontal stretch by a factor of 4 in the plane.

Compute the area of the quadrilateral after applying the horizontal stretch by factor 4 to the vertices $(3,4)$, $(6,4)$, $(6,0)$, $(3,0)$.

Calculate the area of the quadrilateral with vertices $(3,4)$, $(6,4)$, $(6,0)$ and $(3,0)$ using the determinant method.

Explain why the rotation by $90^\circ$ CCW does not change the absolute value of the determinant of the preceding transformation.

Find the coordinates of the image of the point $(6,1)$ under the horizontal stretch by factor 4.

Compute the determinant of the final transformation matrix $\begin{pmatrix}0 & -2\\8 & 0\end{pmatrix}$ and deduce the area of the image of the original quadrilateral.

Compute the determinant of the matrix $\begin{pmatrix}8 & 0\\0 & 2\end{pmatrix}$ and interpret its geometric meaning.

Determine the combined transformation matrix for first applying a horizontal stretch by 4 and then an enlargement by 2.

Determine the image coordinates of the vertex $(3,4)$ after the full transformation (stretch by 4, enlarge by 2, rotate $90^\circ$ CCW).

Find the combined transformation matrix for a horizontal stretch by 4, then an enlargement by 2, followed by a $90^\circ$ CCW rotation.