https://file.aiquickdraw.com/mathbot/file/1724423964638ukxthz27.jpg

The matrix provided is:

\[
\begin{pmatrix}
1 & 2 \\
6 & -6
\end{pmatrix}
\]

To find the inverse of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\), the formula is:

\[
\text{Inverse} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For your matrix:

- \(a = 1\)
- \(b = 2\)
- \(c = 6\)
- \(d = -6\)

First, calculate the determinant (\(\text{det}\)):

\[
\text{det} = ad - bc = (1)(-6) - (2)(6) = -6 - 12 = -18
\]

Since the determinant is not zero, the matrix is invertible. The inverse is:

\[
\text{Inverse} = \frac{1}{-18} \begin{pmatrix} -6 & -2 \\ -6 & 1 \end{pmatrix}
\]

Simplifying:

\[
\text{Inverse} = \begin{pmatrix} \frac{1}{3} & \frac{1}{9} \\ \frac{1}{3} & -\frac{1}{18} \end{pmatrix}
\]

So the inverse of the matrix is:

\[
\begin{pmatrix} \frac{1}{3} & \frac{1}{9} \\ \frac{1}{3} & -\frac{1}{18} \end{pmatrix}
\]

Would you like any more details or have any questions?

Here are some follow-up questions you might consider:
1. How do you verify the inverse of a matrix?
2. What does it mean if the determinant of a matrix is zero?
3. How does matrix inversion apply in solving systems of linear equations?
4. What are the properties of invertible matrices?
5. How can matrix inversion be extended to larger matrices?

**Tip:** Always check the determinant before trying to find the inverse of a matrix; if it's zero, the matrix is not invertible!

Learn how to find the inverse of the matrix 1 2 6 -6 with a detailed step-by-step solution. This problem covers key concepts in matrix algebra, including the formula for finding the inverse of a 2x2 matrix and determinant calculation. Suitable for high school students in Grades 10-12.

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