$A=\left(\begin{array}{cc}-1& 5\\ 2& 2\end{array}\right),B=\left(\begin{array}{cc}3& 2\\ 2& -1\end{array}\right),\text{and}C=\left(\begin{array}{cc}6& 8\\ 3& t\end{array}\right)$
Find:
(a) AB,

(b) The value of t for which C has no inverse.

Given that $\left(\begin{array}{cc}2& 4\\ a& 3\end{array}\right)+k\left(\begin{array}{cc}3& 1\\ 0& -2\end{array}\right)=\left(\begin{array}{cc}8& 6\\ -3& -1\end{array}\right)$.
(a) Find the values of a and k.

(b) Given that $\left(\begin{array}{cc}1& -2\\ x& 4\end{array}\right)$ has no inverse, find the value of x.

$M=\left(\begin{array}{cc}1& 1\\ 1& 2\end{array}\right),M^2=\left(\begin{array}{cc}2& 3\\ 3& 5\end{array}\right)$
(a) Find M³.

(b) $M^5=\left(\begin{array}{cc}x& 55\\ y& 89\end{array}\right)$
Find the value of x and the value of y.

Triangle ABC has vertices A(1, 1), B(3, 1) and C(3, 4).
Triangle ABC is transformed by the matrix $P=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$.
(a) Find the coordinates of B and C after transformation P.

(b) Describe fully the transformation P.

$A=\left(\begin{array}{cc}3& 1\\ 7& x\end{array}\right)$
Calculate the value of x when $\left|A\right|=x+3$.

$A=\left(\begin{array}{cc}-5& 7\\ 3& -4\end{array}\right)$
Find $A^{-1}$.

$A=\left(\begin{array}{cc}4& 2\\ 5& 3\end{array}\right)\text{and}C=\left(\begin{array}{c}2\\ -1\end{array}\right)$
(a) Find the matrix AC.

(b) Find $A^{-1}$.

$A=\left(\begin{array}{cc}4& 2\\ 0& 3\end{array}\right)\text{,}B=\left(\begin{array}{cc}\frac{1}{4}& k\\ 0& \frac{1}{3}\end{array}\right)\text{,}C=\left(\begin{array}{cc}12& 0\\ -9& m\end{array}\right)$
Find:
(a) A²,

(b) k if AB = I,

(c) m if the determinant of A is equal to the determinant of C.

(a) The determinant of the matrix $\left(\begin{array}{cc}x& (x+1)\\ 3x& 4x\end{array}\right)$ is −2.
(i) Use this information to form an equation and show that it can be reduced to $x^2-3x+2=0$.

(ii) Solve $x^2-3x+2=0$.

(b) Find the values of x and y:
$\left(\begin{array}{c}x\\ 2\end{array}\right)+\left(\begin{array}{c}-3\\ 5\end{array}\right)=\left(\begin{array}{c}2\\ y\end{array}\right)$

(a) P is the point (3, 5).
(i) Write the coordinates of the image of P under a translation through $\left(\begin{array}{c}6\\ -4\end{array}\right)$.

(ii) G and M are transformations represented by the matrices $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\left(\begin{array}{cc}1& 0\\ 0& -2\end{array}\right)$, respectively.
Q is the image of P under M, and R is the image of Q under G.
Write down the coordinates of R.

(b) Describe fully a single transformation that is represented by the matrix $\left(\begin{array}{cc}1& -2\\ 0& 1\end{array}\right)$.

(c) State the necessary condition for the existence of a transformation matrix.

(d) $\overrightarrow{A}M$ = $\left(\begin{array}{c}1\\ -6\end{array}\right)$, $\overrightarrow{R}N$ = $\left(\begin{array}{c}-7\\ 4\end{array}\right)$ and $\overrightarrow{N}M$ = $\left(\begin{array}{c}7\\ 9\end{array}\right)$.
(i) Work out $\overrightarrow{A}R$.

(ii) M is the point (1, 0) and $\left|\overrightarrow{M}Q\right|$ = $\left|\overrightarrow{R}N\right|$.
Find two possible coordinates of Q that are positive integers.

$A=\left(\begin{array}{cc}-1& 2\\ 0& 1\end{array}\right)$, $B=\left(\begin{array}{cc}2& 0\\ 1& 4\end{array}\right)$, $C=\left(\begin{array}{cc}2& -3\end{array}\right)$.
Find the following matrices.
(a) CA

(b) B⁻¹

(a) $B=\left(\begin{array}{cc}x& 3\\ 5& 4\end{array}\right)$, and |B| = 3x.
Find the value of x.

(b) $R=\left(\begin{array}{cc}3& 3\\ 5& 8\end{array}\right)$, $D=\left(\begin{array}{c}3\\ -6\end{array}\right)$, $T=\left(\begin{array}{cc}9& 1\end{array}\right)$.
Work out:
(i) TD

(ii) R²

(iii) R⁻¹