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To find the inverse of the matrix \( A = \begin{pmatrix} 1 & 1 \\ -3 & 3 \end{pmatrix} \), we need to follow these steps:

1. **Check if the matrix is invertible**: A matrix is invertible if its determinant is non-zero.
   \[
   \text{det}(A) = (1)(3) - (1)(-3) = 3 + 3 = 6
   \]
   Since the determinant is 6, which is not zero, the matrix is invertible.

2. **Find the inverse formula**: The inverse of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
   \[
   A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
   \]

3. **Apply the formula to our matrix**:
   \[
   A^{-1} = \frac{1}{6} \begin{pmatrix} 3 & -1 \\ 3 & 1 \end{pmatrix}
   \]

4. **Final result**:
   \[
   A^{-1} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{6} \\ \frac{1}{2} & \frac{1}{6} \end{pmatrix}
   \]

This is the inverse of the matrix \( A \).

Would you like any further details or explanations on this process?

Here are 5 questions to expand on the topic:
1. What is the significance of the determinant in determining if a matrix is invertible?
2. How would you find the inverse of a 3x3 matrix?
3. Can every square matrix be inverted? Why or why not?
4. How is the inverse of a matrix applied in solving systems of linear equations?
5. What does it mean for two matrices to be inverses of each other?

**Tip**: Always check the determinant first when determining if a matrix is invertible; if it's zero, the matrix doesn't have an inverse.

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

This article covers the detailed step-by-step solution for finding the inverse of the 2x2 matrix A = [[1, 1], [-3, 3]]. Learn about key linear algebra concepts, including determinants and matrix inversion, and follow through the provided formulas to compute the result. Suitable for college-level students studying linear algebra.

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