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

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Question 1: Given the matrix $Q = \begin{bmatrix} 3 & 2 \\ x & 4 \end{bmatrix}$. Find the value of $x$ given that the determinant of $Q$ is 2.

Step 1: Calculate the determinant of $Q$. For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$. $\det(Q) = (3)(4) - (2)(x) = 12 - 2x$

Step 2: Set the determinant equal to 2 and solve for $x$. $12 - 2x = 2$ $10 = 2x$ $x = \frac{10}{2}$ $x = 5$ The value of $x$ is $\boxed{5}$.

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.

Step 1: Calculate the determinant of $M$. $\det(M) = (3)(x) - (2)(5) = 3x - 10$

Step 2: Set the determinant equal to 23 and solve for $x$. $3x - 10 = 23$ $3x = 23 + 10$ $3x = 33$ $x = \frac{33}{3}$ $x = 11$ The value of $x$ is $\boxed{11}$.

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

Step 1: Substitute the value of $x$ found in part 2a) i) into matrix $M$. From 2a) i), $x=11$. $M = \begin{bmatrix} 3 & 2 \\ 5 & 11 \end{bmatrix}$ The determinant of $M$ is given as 23.

Step 2: Calculate the inverse of $M$. 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.

Step 1: Calculate the determinant of matrix $A$. $\det(A) = (4)(2) - (-5)(1) = 8 - (-5) = 8 + 5 = 13$

Step 2: Calculate the determinant of matrix $B$. $\det(B) = (7)(5) - (y)(3) = 35 - 3y$

Step 3: Set the determinants equal and solve for $y$. $\det(A) = \det(B)$ $13 = 35 - 3y$ $3y = 35 - 13$ $3y = 22$ $y = \frac{22}{3}$ The value of $y$ is $\boxed{\frac{22}{3}}$.

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

Step 1: Substitute the value of $y$ found in part 2b) i) into matrix $B$. 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 $\det(A)=13$).

Step 2: Calculate the inverse of $B$. $B^{-1} = \frac{1}{13} \begin{bmatrix} 5 & -\frac{22}{3} \\ -3 & 7 \end{bmatrix}$ 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 $\boxed{\frac{1}{13} \begin{bmatrix} 5 & -\frac{22}{3} \\ -3 & 7 \end{bmatrix}}$.

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