Find the inverse of the matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \)

Step 1 ：Step 1: Find the determinant of matrix A, denoted as |A|. \( |A| = 1*4 - 2*3 = -2 \)

Step 2 ：Step 2: Find the adjugate of matrix A, which is the transpose of the cofactor matrix. The cofactor matrix is obtained by replacing each element with its cofactor, and the cofactor of an element a[i][j] is (-1)^(i+j)*|M[i][j]|, where M[i][j] is the (i,j) minor of A. The adjugate of A, denoted as adj(A), is \( \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} \)

Step 3 ：Step 3: The inverse of matrix A, denoted as A^(-1), is 1/|A|*adj(A). Therefore, \( A^{-1} = \frac{1}{-2} \begin{bmatrix} 4 & -2 \ -3 & 1 \end{bmatrix} = \begin{bmatrix} -2 & 1 \ 1.5 & -0.5 \end{bmatrix} \)