Matrix Multiplication (AQA GCSE Further Maths): Exam Questions

$\mathbf{A}=\left(\begin{array}{cc}4& -1\\ -7& 2\end{array}\right)$              $\mathbf{B}=\left(\begin{array}{c}s\\ -5\end{array}\right)$               $\mathbf{C}=\left(\begin{array}{c}-1\\ t\end{array}\right)$              $\mathbf{D}=\left(\begin{array}{cc}2& 1\\ 7& u\end{array}\right)$

$s,t$ and $u$ are constants.

$\mathbf{AB}=\mathbf{C}$

Work out the values of $s$ and $t$.

$s=$.....................................

$t=$.....................................

$\mathbf{AD}=\mathbf{I}$

Work out the value of $u$.

$u=$..........................

$\left(\begin{array}{cc}1& k\\ -3& 0\end{array}\right)\left(\begin{array}{cc}0& -k\\ 3& 1\end{array}\right)=12\mathbf{I}$ where $\mathbf{I}$ is the identity matrix.

Find the value of $k$.

$\left(\begin{array}{cc}2& 1\\ 0& 3\end{array}\right)\left(\begin{array}{cc}a& b\\ 0& 0.4\end{array}\right)=k\mathbf{I}$ where $k$ is a constant and $\mathbf{I}$ is the identity matrix.

Work out the values of $a$ and $b$.

$a=.................b=.................$   

$\mathbf{M}=\left(\begin{array}{cc}4& -5\\ -1& 2\end{array}\right)$ and $\mathbf{P}=\left(\begin{array}{cc}7& a\\ b& c\end{array}\right)$

${\mathbf{M}}^2=k\mathbf{P}$ where ${\mathbf{M}}^2=\mathbf{MM}$ and $k$ is an integer.

Find the values of $a$, $b$ and $c$.

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

$\mathbf{C}=\mathbf{AB}$

$\mathbf{D}=\mathbf{AC}$

If $\mathbf{D}=\left(\begin{array}{c}kq\\ 0\end{array}\right)$ where $k$ is an integer, find the value of $k$.

$\mathbf{A}=\left(\begin{array}{cc}1& 4\\ 5& 2\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{cc}-4& p\\ q& -2\end{array}\right)$ where $p$ and $q$ are integers.

If $\mathbf{AB}=\mathbf{BA}$, find the values of $p$ and $q$.

$\left(\begin{array}{cc}x& 5-x^2\\ -3& -1\end{array}\right)\left(\begin{array}{cc}x-2& 1\\ 2& 1\end{array}\right)=\left(\begin{array}{cc}2& a\\ b& c\end{array}\right)$ where $x$ is negative and $a$, $b$ and $c$ are integers.

Find the values of $a$, $b$ and $c$.

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

Find the possible values of $x$ and $y$.

Do not use trial and improvement.

You must show your working.