https://file.aiquickdraw.com/mathbot/file/172746235728979ifs92b.jpg

The matrix in question is:

\[
A = \begin{pmatrix} 1 & 1 \\ -3 & 3 \end{pmatrix}
\]

To find the inverse of a 2x2 matrix, we can use the formula:

\[
A^{-1} = \frac{1}{\text{det}(A)} \cdot \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

Where the original matrix is:

\[
A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]

So, for the given matrix \(A = \begin{pmatrix} 1 & 1 \\ -3 & 3 \end{pmatrix}\):

- \(a = 1\)
- \(b = 1\)
- \(c = -3\)
- \(d = 3\)

### Step 1: Calculate the determinant of \(A\)
The determinant of a 2x2 matrix is given by:

\[
\text{det}(A) = ad - bc
\]

Substituting the values:

\[
\text{det}(A) = (1)(3) - (1)(-3) = 3 + 3 = 6
\]

### Step 2: Apply the inverse formula
Now, apply the formula for the inverse:

\[
A^{-1} = \frac{1}{6} \cdot \begin{pmatrix} 3 & -1 \\ 3 & 1 \end{pmatrix}
\]

Multiplying the matrix by \(\frac{1}{6}\):

\[
A^{-1} = \begin{pmatrix} \frac{3}{6} & \frac{-1}{6} \\ \frac{3}{6} & \frac{1}{6} \end{pmatrix}
\]

This simplifies to:

\[
A^{-1} = \begin{pmatrix} \frac{1}{2} & \frac{-1}{6} \\ \frac{1}{2} & \frac{1}{6} \end{pmatrix}
\]

Thus, the inverse of the matrix \(A\) is:

\[
A^{-1} = \begin{pmatrix} \frac{1}{2} & \frac{-1}{6} \\ \frac{1}{2} & \frac{1}{6} \end{pmatrix}
\]

Let me know if you would like any additional details or further explanation!

### Related Questions:
1. What is the determinant of a matrix and why is it important when finding an inverse?
2. How do we know if a matrix is invertible?
3. What happens if the determinant of a matrix is zero?
4. Can you explain the geometric interpretation of an inverse matrix?
5. How does the inverse matrix change the system of equations when solving for variables?

### Tip:
Always check if the determinant of the matrix is non-zero before attempting to find the inverse; if the determinant is zero, the matrix is singular and does not have an inverse.

Find the inverse of the matrix A = [[1, 1], [-3, 3]].

Learn how to find the inverse of a 2x2 matrix A = [[1, 1], [-3, 3]] by calculating the determinant and applying the matrix inversion formula. This detailed solution is suitable for high school and college students studying linear algebra.

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