To find the inverse of a 3x3 matrix, you can use the following steps, provided the matrix is invertible (i.e., its determinant is non-zero):

1. Find the Determinant: Calculate the determinant $\text{det}(A)$ of the 3x3 matrix $A$. For a matrix $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$ the determinant is calculated as: $\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

2. Calculate the Matrix of Minors: For each element in the matrix, calculate the determinant of the 2x2 submatrix that remains after removing the row and column of that element. This will give you a new 3x3 matrix, known as the matrix of minors.

3. Form the Cofactor Matrix: Apply a checkerboard pattern of signs (+, -, + in the first row, etc.) to the matrix of minors to create the cofactor matrix.

4. Transpose the Cofactor Matrix: This gives the adjugate (or adjoint) matrix.

5. Divide by the Determinant: Finally, multiply each element in the adjugate matrix by ( \frac{1}{\text{