Problem
Given the matrix \( A = \begin{bmatrix} 3 & 2 \\ 4 & 1 \end{bmatrix} \), calculate the inverse of matrix A.
Solution
Step 1 ：First, we calculate the determinant of the matrix A: \( det(A) = 3*1 - 2*4 = -5 \).
Step 2 ：Next, we swap the elements on the main diagonal, so we get a new matrix \( B = \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \).
Step 3 ：Then, we change the signs of the elements on the other diagonal, giving us matrix \( C = \begin{bmatrix} 1 & -2 \\ -4 & 3 \end{bmatrix} \).
Step 4 ：Finally, we divide each element of matrix C by det(A) to get the inverse of A: \( A^{-1} = \frac{1}{-5} * \begin{bmatrix} 1 & -2 \\ -4 & 3 \end{bmatrix} = \begin{bmatrix} -0.2 & 0.4 \\ 0.8 & -0.6 \end{bmatrix} \).
