Problem
If \( A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \), find the matrix \( B \) such that \( AB = I \) where \( I \) is the identity matrix.
Solution
Step 1 ：Step 1: Start with the equation \( AB = I \).
Step 2 ：Step 2: To solve for \( B \), we need to find the inverse of \( A \) because \( A^{-1}A = I \). So, \( B = A^{-1} \).
Step 3 ：Step 3: The formula for the inverse of a 2x2 matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is \( A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
Step 4 ：Step 4: Substituting the given values, we get \( A^{-1} = \frac{1}{2*5 - 3*4} \begin{bmatrix} 5 & -3 \\ -4 & 2 \end{bmatrix} = \frac{1}{-2} \begin{bmatrix} 5 & -3 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} -5/2 & 3/2 \\ 2 & -1 \end{bmatrix} \).
