Find the inverse of the matrix \( A = \begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix} \)
Answer
Step 4: Divide each element of the matrix by the determinant to get \( A^{-1} \): \( A^{-1} = \frac{1}{7} \begin{bmatrix} 5 & -4 \\ -2 & 3 \end{bmatrix} \)
Step 1 ：Step 1: Find the determinant (ad - bc) of the matrix \( A \). Where a = 3, b = 4, c = 2, and d = 5. Therefore, \( det(A) = (3*5) - (4*2) = 7 \)
Step 2 ：Step 2: Swap the elements of the main diagonal (a and d). Hence, \( a = 5 \) and \( d = 3 \)
Step 3 ：Step 3: Change the signs of the elements off the main diagonal (b and c). Hence, \( b = -4 \) and \( c = -2 \)
Step 4 ：Step 4: Divide each element of the matrix by the determinant to get \( A^{-1} \): \( A^{-1} = \frac{1}{7} \begin{bmatrix} 5 & -4 \\ -2 & 3 \end{bmatrix} \)
