Given the matrix Q = 3 & 2 x & 4 . Find the value of x given that the determinant of Q is 2.

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$.

$$\det(Q) = (3)(4) - (2)(x) = 12 - 2x$$

$$12 - 2x = 2$$ $$10 = 2x$$ $$x = \frac{10}{2}$$ $$x = 5$$

The value of $x$ is $\boxed{\text{5}}$.

Question 2a) i):
Given the matrix $M = \begin{bmatrix} 3 & 2 \\ 5 & x \end{bmatrix}$. Find the value of $x$ for which the determinant is 23.

$$3x - 10 = 23$$ $$3x = 23 + 10$$ $$3x = 33$$ $$x = \frac{33}{3}$$ $$x = 11$$

The value of $x$ is $\boxed{\text{11}}$.

Question 2a) ii):
Hence find the inverse of $M$.

From 2a) i), $x=11$.

$$M = \begin{bmatrix} 3 & 2 \\ 5 & 11 \end{bmatrix}$$

The determinant of $M$ is given as 23.

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the inverse is $\frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.

$$M^{-1} = \frac{1}{23} \begin{bmatrix} 11 & -2 \\ -5 & 3 \end{bmatrix}$$

The inverse of $M$ is $\boxed{\frac{1}{23} \begin{bmatrix} 11 & -2 \\ -5 & 3 \end{bmatrix}}$.

Question 2b) i):
Given that Matrix $A = \begin{bmatrix} 4 & -5 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 7 & y \\ 3 & 5 \end{bmatrix}$. Find the value of $y$ for which the determinant of $A$ and $B$ are equal.

$$\det(A) = \det(B)$$ $$13 = 35 - 3y$$ $$3y = 35 - 13$$ $$3y = 22$$ $$y = \frac{22}{3}$$

The value of $y$ is $\boxed{\frac{22}{3}}$.

Question 2b) ii):
Hence find the inverse of $B$.

From 2b) i), $y=\frac{22}{3}$.

$$B = \begin{bmatrix} 7 & \frac{22}{3} \\ 3 & 5 \end{bmatrix}$$

The determinant of $B$ is 13 (since $\det(A) = \det(B)$ and $\det(A)=13$).

$$B^{-1} = \frac{1}{13} \begin{bmatrix} 5 & -\frac{22}{3} \\ -3 & 7 \end{bmatrix}$$

You can also distribute the $\frac{1}{13}$:

$$B^{-1} = \begin{bmatrix} \frac{5}{13} & -\frac{22}{39} \\ -\frac{3}{13} & \frac{7}{13} \end{bmatrix}$$

The inverse of $B$ is $\boxed{\frac{1}{13} \begin{bmatrix} 5 & -\frac{22}{3} \\ -3 & 7 \end{bmatrix}}$.