The problem requires obtaining the inverse of the matrix

$\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix}$

and using it to solve the system of equations:

$x + 3y = 4 \quad \text{and} \quad 2x - 2y = 9$

Step 1: Finding the Inverse of the Matrix

For a 2x2 matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the inverse is given by:

$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

In this case, the matrix is:

$\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix}$

So, we have:

• $a = 1$
• $b = 3$
• $c = 2$
• $d = -2$

The determinant $\text{det}(A) = ad - bc$ is:

$\text{det}(A) = (1)(-2) - (3)(2) = -2 - 6 = -8$

Now, the inverse of the matrix is:

$A^{-1} = \frac{1}{-8} \begin{pmatrix} -2 & -3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix}$

Step 2: Solving the System of Equations

We can express the system of equations as:

$\begin{pmatrix} 1 & 3 \\ 2 & -2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 9 \end{pmatrix}$

Now, multiply both sides of the equation by the inverse matrix $A^{-1}$:

$\begin{pmatrix} x \\ y \end{pmatrix} = A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix}$

First, compute:

$A^{-1} \begin{pmatrix} 4 \\ 9 \end{pmatrix} = \begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix} \begin{pmatrix} 4 \\ 9 \end{pmatrix}$

This gives:

$\begin{pmatrix} \frac{1}{4}(4) + \frac{3}{8}(9) \\ \frac{1}{4}(4) - \frac{1}{8}(9) \end{pmatrix} = \begin{pmatrix} 1 + \frac{27}{8} \\ 1 - \frac{9}{8} \end{pmatrix} = \begin{pmatrix} \frac{35}{8} \\ \frac{-1}{8} \end{pmatrix}$

Thus, the solution is:

$x = \frac{35}{8}, \quad y = -\frac{1}{8}$

Final Answer:

• The inverse of the matrix is $\begin{pmatrix} \frac{1}{4} & \frac{3}{8} \\ \frac{1}{4} & -\frac{1}{8} \end{pmatrix}$.
• The solution to the system of equations is $x = \frac{35}{8}$ and $y = -\frac{1}{8}$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What are the conditions for a 2x2 matrix to have an inverse?
2. How do we calculate the determinant of a 3x3 matrix?
3. Can a matrix with a determinant of zero have an inverse?
4. How can you solve a system of equations using Gaussian elimination?
5. How does Cramer's rule work in solving a system of equations?

Tip: The determinant of a matrix gives us key information about whether it is invertible. A determinant of zero means the matrix is singular and has no inverse.