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

Determine the image coordinates of the vertex $(3,4)$ after the full transformation (horizontal stretch with scale factor 4, enlargement with scale factor 2, rotation of $90^\circ$ counter-clockwise about the origin).

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

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

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

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 $90^\circ$ anticlockwise does not change the absolute value of the determinant of the preceding transformation.

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

A quadrilateral $Q$ has an area of $12\text{ units}^2$. It is transformed by the matrix

$\boldsymbol{M} = \begin{pmatrix} 0 & -2 \\ 8 & 0 \end{pmatrix}$

to form an image $Q'$.

Calculate the determinant of $\boldsymbol{M}$ and deduce the area of $Q'$.

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

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

Find the combined transformation matrix for a horizontal stretch by 4, then an enlargement by a scale factor of 2, followed by a $90^\circ$ counter-clockwise rotation about the origin.

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