If matrix \( A = \begin{bmatrix} 1 & 2 \cr 3 & 4 \end{bmatrix} \), what is \( A^{-1} \), the inverse of matrix A?
Answer
Step 4: Divide all elements of the matrix by the determinant. The resulting matrix is \( \begin{bmatrix} -2 & 1 \cr 1.5 & -0.5 \end{bmatrix} \), which is the inverse of matrix A.
Step 1 ：Step 1: Calculate the determinant of the matrix, denoted as \( det(A) \). \( det(A) = 1\times4 - 2\times3 = 4 - 6 = -2 \).
Step 2 ：Step 2: Swap the elements of the main diagonal. The matrix becomes \( \begin{bmatrix} 4 & 2 \cr 3 & 1 \end{bmatrix} \).
Step 3 ：Step 3: Change the sign of the elements of the secondary diagonal. The matrix becomes \( \begin{bmatrix} 4 & -2 \cr -3 & 1 \end{bmatrix} \).
Step 4 ：Step 4: Divide all elements of the matrix by the determinant. The resulting matrix is \( \begin{bmatrix} -2 & 1 \cr 1.5 & -0.5 \end{bmatrix} \), which is the inverse of matrix A.
