https://file.aiquickdraw.com/mathbot/file/17242474080742tjlq4ni.jpg

The matrix you've provided is:

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

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

\[
\text{If } A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \text{ then } A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}
\]

For the given matrix:

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

Here, \( a = 1 \), \( b = 1 \), \( c = 1 \), and \( d = -2 \).

First, we calculate the determinant \( \text{det}(A) = ad - bc \):

\[
\text{det}(A) = (1)(-2) - (1)(1) = -2 - 1 = -3
\]

Since the determinant is non-zero, the matrix is invertible. Now we can find the inverse:

\[
A^{-1} = \frac{1}{-3} \begin{pmatrix} -2 & -1 \\ -1 & 1 \end{pmatrix}
\]

Simplifying further:

\[
A^{-1} = \begin{pmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & -\frac{1}{3} \end{pmatrix}
\]

So, the inverse of the matrix is:

\[
\boxed{A^{-1} = \begin{pmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{3} & -\frac{1}{3} \end{pmatrix}}
\]

Would you like more details or have any questions?

Here are some related questions to consider:
1. How is the determinant of a matrix calculated for larger matrices?
2. What happens if the determinant of a matrix is zero?
3. How can you verify that a matrix and its inverse multiply to give the identity matrix?
4. What are the applications of matrix inverses in solving systems of linear equations?
5. How would you find the inverse of a 3x3 matrix?

**Tip:** When calculating the determinant of larger matrices, remember to break them down into smaller matrices using cofactor expansion.

Learn how to find the inverse of the matrix 1 1 1 -2 with a step-by-step solution. This problem covers matrix algebra concepts, including the formula for the inverse of a 2x2 matrix and the determinant calculation. Suitable for college-level mathematics students.

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