https://r2.erweima.ai/mathbot/image/1724716528321a6ry71lm.jpg

The image shows a matrix:

\[
\begin{pmatrix}
1 & 1 \\
3 & 4
\end{pmatrix}
\]

To find the identity matrix, you need to understand that the identity matrix is a square matrix with ones on the main diagonal and zeros elsewhere. However, you might be asking for the inverse of the given matrix, as the identity matrix is generally the result of multiplying a matrix by its inverse.

To find the inverse of a \(2 \times 2\) matrix, the formula is:

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

For the matrix \( A = \begin{pmatrix} 1 & 1 \\ 3 & 4 \end{pmatrix} \):

1. **Calculate the determinant** (\(\text{det}(A)\)):

   \[
   \text{det}(A) = (1 \times 4) - (1 \times 3) = 4 - 3 = 1
   \]

2. **Apply the inverse formula**:

   \[
   A^{-1} = \frac{1}{1} \cdot \begin{pmatrix} 4 & -1 \\ -3 & 1 \end{pmatrix} = \begin{pmatrix} 4 & -1 \\ -3 & 1 \end{pmatrix}
   \]

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

\[
\begin{pmatrix} 4 & -1 \\ -3 & 1 \end{pmatrix}
\]

Multiplying the original matrix by its inverse will yield the identity matrix.

Do you want more details on the steps, or do you have any questions?

Here are some related questions for further exploration:
1. What is the general method to find the inverse of a \(3 \times 3\) matrix?
2. How can determinants be used to solve systems of linear equations?
3. What are the properties of the identity matrix in different dimensions?
4. What happens if the determinant of a matrix is zero?
5. How can you verify that two matrices are inverses of each other?

**Tip:** When working with matrices, always check the determinant before attempting to find the inverse; if it's zero, the matrix does not have an inverse!

Learn how to find the inverse of a 2x2 matrix with a step-by-step guide. This tutorial covers the formula for finding the inverse, calculating the determinant, and verifying the result. Suitable for advanced high school students and anyone interested in linear algebra.

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