Transformations using Matrices (Edexcel A Level Further Maths: Core Pure): Exam Questions

$\mathbf{A}=\left(\begin{array}{cc}2& a\\ a-4& b\end{array}\right)$

where $a$ and $b$ are non-zero constants.

Given that the matrix $\mathbf{A}$ is self-inverse,

determine the value of $b$ and the possible values for $a$.

The matrix $\mathbf{A}$ represents a linear transformation $M$.

Using the smaller value of $a$ from part (a),

show that the invariant points of the linear transformation $M$ form a line, stating the equation of this line.

$\mathbf{P}=\left(\begin{array}{cc}p& 2p\\ -1& 3p\end{array}\right)$

where $p$ is a positive constant.

The matrix $\mathbf{P}$ represents a linear transformation $U$.

The triangle $T$ has vertices at the points with coordinates (1, 2), (3, 2) and (2, 5).

The area of the image of $T$ under the linear transformation $U$ is 15

Determine the value of $p$.

The transformation $V$ consists of a stretch scale factor 3 parallel to the $x$-axis with the $y$-axis invariant followed by a stretch scale factor –2 parallel to the $y$-axis with the $x$-axis invariant. The transformation $V$ is represented by the matrix $\mathbf{Q}$.

Write down the matrix $\mathbf{Q}$.

Given that $U$ followed by $V$ is the transformation $W$, which is represented by the matrix $\mathbf{R}$,

find the matrix $\mathbf{R}$.

$\mathbf{M}=\left(\begin{array}{cc}4& -5\\ 2& -7\end{array}\right)$

Show that the matrix $\mathbf{M}$ is non-singular.

The transformation $T$ of the plane is represented by the matrix $\mathbf{M}$.

The triangle $R$ is transformed to the triangle $S$ by the transformation $T$.

Given that the area of $S$ is 63 square units,

find the area of $R.$

Show that the line $y=2x$ is invariant under the transformation $T.$