1. Compute the inverse of the matrix $$\begin{bmatrix} 5 & 3 \\ 2 & 6 \end{bmatrix}$$ 2. Use the inverse matrix method to solve the following: 2p-4q = -14 -2q-3p = 13 3. Descr... 1. Compute the inverse of the matrix $$\begin{bmatrix} 5 & 3 \\ 2 & 6 \end{bmatrix}$$ 2. Use the inverse matrix method to solve the following: 2p-4q = -14 -2q-3p = 13 3. Describe each of the following types of sets. Read more

Steps to Solve

1. Compute the determinant of the matrix

Let the given matrix be $A = \begin{bmatrix} 5 & 3 \ 2 & 6 \end{bmatrix}$. The determinant of $A$ is given by: $det(A) = (5 \times 6) - (3 \times 2) = 30 - 6 = 24$

2. Find the adjugate (adjoint) of the matrix

The adjugate of $A$ is obtained by swapping the diagonal elements and changing the sign of the off-diagonal elements. $adj(A) = \begin{bmatrix} 6 & -3 \ -2 & 5 \end{bmatrix}$

3. Compute the inverse of the matrix

The inverse of $A$ is given by: $A^{-1} = \frac{1}{det(A)} \times adj(A)$ $A^{-1} = \frac{1}{24} \begin{bmatrix} 6 & -3 \ -2 & 5 \end{bmatrix} = \begin{bmatrix} \frac{6}{24} & \frac{-3}{24} \ \frac{-2}{24} & \frac{5}{24} \end{bmatrix} = \begin{bmatrix} \frac{1}{4} & -\frac{1}{8} \ -\frac{1}{12} & \frac{5}{24} \end{bmatrix}$

4. Rewrite the system of equations in matrix form

The given system of equations is: $2p - 4q = -14$ $-3p - 2q = 13$ Representing this system in matrix form $AX = B$ where $A$ is the coefficient matrix, $X$ is the variable matrix, and $B$ is the constant matrix. $A = \begin{bmatrix} 2 & -4 \ -3 & -2 \end{bmatrix}, X = \begin{bmatrix} p \ q \end{bmatrix}, B = \begin{bmatrix} -14 \ 13 \end{bmatrix}$

5. Calculate the inverse of the coefficient matrix

$det(A) = (2 \times -2) - (-4 \times -3) = -4 - 12 = -16$ $adj(A) = \begin{bmatrix} -2 & 4 \ 3 & 2 \end{bmatrix}$ $A^{-1} = \frac{1}{-16} \begin{bmatrix} -2 & 4 \ 3 & 2 \end{bmatrix} = \begin{bmatrix} \frac{1}{8} & -\frac{1}{4} \ -\frac{3}{16} & -\frac{1}{8} \end{bmatrix}$

6. Solve for the variables using the inverse matrix method

$X = A^{-1}B$ $\begin{bmatrix} p \ q \end{bmatrix} = \begin{bmatrix} \frac{1}{8} & -\frac{1}{4} \ -\frac{3}{16} & -\frac{1}{8} \end{bmatrix} \begin{bmatrix} -14 \ 13 \end{bmatrix}$ $\begin{bmatrix} p \ q \end{bmatrix} = \begin{bmatrix} (\frac{1}{8} \times -14) + (-\frac{1}{4} \times 13) \ (-\frac{3}{16} \times -14) + (-\frac{1}{8} \times 13) \end{bmatrix}$ $\begin{bmatrix} p \ q \end{bmatrix} = \begin{bmatrix} -\frac{14}{8} - \frac{13}{4} \ \frac{42}{16} - \frac{13}{8} \end{bmatrix} = \begin{bmatrix} -\frac{7}{4} - \frac{13}{4} \ \frac{21}{8} - \frac{13}{8} \end{bmatrix} = \begin{bmatrix} -\frac{20}{4} \ \frac{8}{8} \end{bmatrix} = \begin{bmatrix} -5 \ 1 \end{bmatrix}$ Therefore, $p = -5$ and $q = 1$

7. Describe types of sets

Since no specific sets are mentioned for description, will describe 3 common ones

• Empty Set: A set containing no elements. Denoted by $ {} $ or $ \emptyset $.
• Finite Set: A set with a countable number of elements. It is possible to list all elements of the set. For example, $ {1, 2, 3, 4} $.
• Infinite Set: A set with an infinite number of elements. It is not possible to list all elements of the set. For example, the set of all natural numbers $ {1, 2, 3, ...} $.