Matrix Algebra (Edexcel International A Level (IAL) Further Maths: Further Pure 1): Exam Questions

(i) The matrix A is defined by

$\mathbf{A}=\left(\begin{array}{cc}3k& 4k-1\\ 2& 6\end{array}\right)$

where $k$ is a constant.

(a) Determine the value of $k$ for which A is singular.

Given that A is non-singular,

[2]

(b) determine ${\mathbf{A}}^{-1}$ in terms of $k$, giving your answer in simplest form.

[2]

Prove by induction that for $n\in {\mathrm{\mathbb{Z}}}^{+}$

${\left(\begin{array}{cc}1& r\\ 0& 2\end{array}\right)}^n=\left(\begin{array}{cc}1& (2^n-1)r\\ 0& 2^n\end{array}\right)$

where $r$ is a constant.

$\mathbf{M}=\left(\begin{array}{cc}4& 0\\ 0& 5\end{array}\right)$ $\mathbf{N}={\left(\begin{array}{cc}1& -2\\ 0& 2\end{array}\right)}^4$

The transformation represented by matrix M followed by the transformation represented by matrix N is represented by the matrix B

(i) Determine N in the form $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ where $a$, $b$, $c$ and $d$ are integers.

[1]

(ii) Determine B

[2]

Hexagon $S$ is transformed onto hexagon $S\text{'}$ by matrix B

Given that the area of $S\text{'}$ is 720 square units, determine the area of $S$

$\mathbf{M}=\left(\begin{array}{cc}2k+1& k\\ k+7& k+4\end{array}\right)$ where $k$ is a constant

Show that M is non-singular for all real values of $k$.

Determine ${\mathbf{M}}^{-1}$ in terms of $k$.

$\mathbf{B}=\left(\begin{array}{cc}a& -4\\ 2& 3\end{array}\right)$ $\mathbf{BC}=\left(\begin{array}{ccc}2& 5& 1\\ 1& 4& 2\end{array}\right)$ where $a$ is a constant

Determine, in terms of $a$, the matrix $\mathbf{C}$

Given that

$\mathbf{A}=\left(\begin{array}{ccc}2& -1& 3\\ -2& 3& 0\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{cc}1& k\\ 0& -3\\ 2k& 2\end{array}\right)$

where $k$ is a non-zero constant,

determine the matrix AB

determine the value of $k$ for which det(AB) = 0

$\mathbf{M}=\left(\begin{array}{cc}k& k\\ 3& 5\end{array}\right)$ where $k$ is a non-zero constant

Determine ${\mathbf{M}}^{-1}$ , giving your answer in simplest form in terms of $k$ .

Hence, given that ${\mathbf{N}}^{-1}=\left(\begin{array}{cc}k& k\\ 4& -1\end{array}\right)$

determine $(\mathbf{MN}{)}^{-1}$, giving your answer in simplest form in terms of $k$.

$\mathbf{A}=\left(\begin{array}{cc}-\frac{\sqrt{3}}{2}& -\frac{1}{2}\\ \frac{1}{2}& -\frac{\sqrt{3}}{2}\end{array}\right)$

Determine the matrix ${\mathbf{A}}^2$

Describe fully the single geometrical transformation represented by the matrix ${\mathbf{A}}^2$

Hence determine the smallest positive integer value of $n$ for which ${\mathbf{A}}^n=\mathbf{I}$

The matrix B represents a stretch scale factor 4 parallel to the $x$-axis.

Write down the matrix B

The transformation represented by matrix A followed by the transformation represented by matrix B is represented by the matrix C

Determine the matrix C

The parallelogram $P$ is transformed onto the parallelogram $P\text{'}$ by the matrix C

Given that the area of parallelogram $P\text{'}$ is 20 square units, determine the area of parallelogram $P$

$\mathbf{M}=\left(\begin{array}{cc}3x& 7\\ 4x+1& 2-x\end{array}\right)$

Find the range of values of $x$ for which the determinant of the matrix $\mathbf{M}$ is positive.

$A=\left(\begin{array}{cc}3& a\\ -2& -2\end{array}\right)$

where $a$ is a non-zero constant and $a\ne 3$

Determine $A^{-1}$ giving your answer in terms of $a$

Given that $A+A^{-1}=I$ where $I$ is the $2\times 2$ identity matrix,

determine the value of $a$

The elements of each matrix should be expressed in exact numerical form.

(a) Write down the $2\times 2$ matrix that represents a rotation of $210°$ anticlockwise about the origin.

[1]

(b) Write down the $2\times 2$ matrix that represents a stretch parallel to the $y$-axis with scale factor 5.

[1]

The transformation $T$ is a rotation of $210°$ anticlockwise about the origin followed by a stretch parallel to the $y$-axis with scale factor 5.

(c) Determine the $2\times 2$ matrix that represents $T$.

[2]

$\mathit{M}=\left(\begin{array}{cc}k& k+3\\ -5& 1-k\end{array}\right)\text{where }k\text{ is a constant}$

(a) Find det $\mathit{M}$, giving your answer in simplest form in terms of $k$.

[2]

A closed shape $R$ is transformed to a closed shape $R\text{'}$ by the transformation represented by the matrix $\mathit{M}$.

Given that the area of $R$ is 2 square units and that the area of $R\text{'}$ is $16k$ square units,

(b) determine the possible values of $k$.

[3]

The matrix M is defined by

$\mathbf{M}=\left(\begin{array}{cc}k+5& -2\\ -3& k\end{array}\right)$

Determine the values of $k$ for which M is singular.

Given that M is non-singular,

find ${\mathbf{M}}^{-1}$ in terms of $k$.