Matrix Algebra (Edexcel IAL Further Maths): Exam Questions

Given that

$A = ( \begin{gathered} k 3 \\ - 1 & k + 2 \end{gathered} ) , \text{where } k \text{ \text{is} a constant}$

show that $\mathrm{det}(A)>0$ for all real values of equation$k$,

find equation$A^{-1}$ in terms of $k$.

(i) The matrix A is defined by

$A = ( \begin{gathered} 3 k & 4 k - 1 \\ 2 6 \end{gathered} )$

where $k$ is a constant.

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

Given that A is non-singular,

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

Prove by induction that for $n\inℤ^{+}$

$( \begin{gathered} 1 r \\ 0 2 \end{gathered} )^{n} = ( \begin{gathered} 1 ( 2^{n} - 1 ) r \\ 0 2^{n} \end{gathered} )$

where $r$ is a constant.

$M = ( \begin{gathered} 4 0 \\ 0 5 \end{gathered} )$ $N = ( \begin{gathered} 1 - 2 \\ 0 2 \end{gathered} )^{4}$

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 $( \begin{gathered} a b \\ c d \end{gathered} )$ where $a$, $b$, $c$ and $d$ are integers.

(ii) Determine B

Hexagon $S$ is transformed onto hexagon equation$S'$ by matrix B

Given that the area of equation$S'$ is 720 square units, determine the area of $S$

$M = ( \begin{gathered} 2 k + 1 & k \\ k + 7 & k + 4 \end{gathered} )$ where equation$k$ is a constant

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

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

$B = ( \begin{gathered} a - 4 \\ 2 3 \end{gathered} )$ $\text{BC} = ( \begin{gathered} 2 5 1 \\ 1 4 2 \end{gathered} )$ where $a$ is a constant

Determine, in terms of $a$, the matrix $C$

Given that

$A = ( \begin{gathered} 2 - 1 & 3 \\ - 2 & 3 0 \end{gathered} )$ and $B = ( \begin{gathered} 1 k \\ 0 - 3 \\ 2 k & 2 \end{gathered} )$

where equation$k$ is a non-zero constant,

determine the matrix AB

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

$M = ( \begin{gathered} k k \\ 3 5 \end{gathered} )$ where $k$ is a non-zero constant

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

Hence, given that $N^{- 1} = ( \begin{gathered} k k \\ 4 - 1 \end{gathered} )$

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

$A = ( \begin{gathered} - \frac{ \sqrt{3} }{2} & - 1 \frac{1}{2} \\ 1 \frac{1}{2} & - \frac{ \sqrt{3} }{2} \end{gathered} )$

Determine the matrix $A^{2}$

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

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

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 $P'$ by the matrix C

Given that the area of parallelogram $P'$ is 20 square units, determine the area of parallelogram $P$

$M = ( \begin{gathered} 3 x & 7 \\ 4 x + 1 & 2 - x \end{gathered} )$

Find the range of values of $x$ for which the determinant of the matrix $M$ is positive.

$A = ( \begin{gathered} 3 a \\ - 2 & - 2 \end{gathered} )$

where $a$ is a non-zero constant and $a\neq3$

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

Given that $A+A^{-1}=I$ where $I$ is the $2\times2$ identity matrix,

determine the value of $a$

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

(a) Write down the $2\times2$ matrix that represents a rotation of $210^{\circ}$ anticlockwise about the origin.

(b) Write down the $2\times2$ matrix that represents a stretch parallel to the $y$-axis with scale factor 5.

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

(c) Determine the $2\times2$ matrix that represents $T$.

$M = ( \begin{gathered} k k + 3 \\ - 5 & 1 - k \end{gathered} ) \text{where } k \text{ \text{is} a constant}$

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

A closed shape $R$ is transformed to a closed shape $R'$ by the transformation represented by the matrix $M$.

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

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

The matrix M is defined by

$M = ( \begin{gathered} k + 5 & - 2 \\ - 3 & k \end{gathered} )$

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

Given that M is non-singular,

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