Find the inverse of the matrix \[ A = \begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix} \]

Step 1 ：The given matrix is \[ A = \begin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix} \]

Step 2 ：The formula to find the inverse of a 2x2 matrix \[ A = \begin{pmatrix} a & b \ c & d \end{pmatrix} \] is \[ A^{-1} = \frac{1}{ad-bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} \]

Step 3 ：We substitute the values a=2, b=3, c=4, d=5 into the formula to get \[ A^{-1} = \frac{1}{2*5 - 3*4} \begin{pmatrix} 5 & -3 \ -4 & 2 \end{pmatrix} \]

Step 4 ：Solving for the determinant, ad-bc, we get -2. Substituting this into the formula, we get \[ A^{-1} = \frac{1}{-2} \begin{pmatrix} 5 & -3 \ -4 & 2 \end{pmatrix} \]

Step 5 ：Finally, multiplying the scalar \(\frac{1}{-2}\) with the matrix, we get the inverse matrix \[ A^{-1} = \begin{pmatrix} -5/2 & 3/2 \ 2 & -1 \end{pmatrix} \]