apply simple method to find the inverse of matrix [■(3&2@4&-1)].

To find the inverse of the \(2 \times 2\) matrix \(\begin{bmatrix} 3 & 2 \\ 4 & -1 \end{bmatrix}\), you can use the simple formula for the inverse of a \(2 \times 2\) matrix.

For a matrix \(\mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), the inverse \(\mathbf{A}^{-1}\) is given by:

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

where \(\text{det}(\mathbf{A}) = ad - bc\) is the determinant of \(\mathbf{A}\).

### Steps:

1. **Calculate the determinant**:
   \[
   \text{det}(\mathbf{A}) = (3 \times -1) - (2 \times 4) = -3 - 8 = -11
   \]

2. **Apply the formula**:
   \[
   \mathbf{A}^{-1} = \frac{1}{-11} \cdot \begin{bmatrix} -1 & -2 \\ -4 & 3 \end{bmatrix}
   \]

3. **Multiply by the scalar**:
   \[
   \mathbf{A}^{-1} = \begin{bmatrix} \frac{1}{11} & \frac{2}{11} \\ \frac{4}{11} & -\frac{3}{11} \end{bmatrix}
   \]

So, the inverse of the matrix \(\begin{bmatrix} 3 & 2 \\ 4 & -1 \end{bmatrix}\) is:

\[
\mathbf{A}^{-1} = \begin{bmatrix} \frac{1}{11} & \frac{2}{11} \\ \frac{4}{11} & -\frac{3}{11} \end{bmatrix}
\]

Would you like details or have any questions about this process?

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Here are some related questions you might find interesting:

1. How do you calculate the determinant of a \(3 \times 3\) matrix?
2. What are the properties of an invertible matrix?
3. How can you check if a matrix is singular or non-singular?
4. What is the role of the determinant in matrix inversion?
5. How does the inverse of a matrix relate to solving systems of linear equations?

**Tip:** Always check the determinant before trying to find the inverse of a matrix. If the determinant is zero, the matrix is not invertible!

Learn how to find the inverse of the 2x2 matrix 3 2 4 -1 with a step-by-step guide. This problem covers matrix algebra concepts, including the formula for the inverse of a 2x2 matrix. Suitable for undergraduate students studying linear algebra.

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