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!