To find the inverse of a 3x3 matrix, you can use the method called "Gaussian elimination" or "row operations." Here are the steps to follow:

  1. Start by writing the given 3x3 matrix as [A|I], where A is the original matrix and I is the identity matrix.

  2. Apply row operations to transform the matrix [A|I] into [I|B], where B is the inverse matrix of A. The goal is to reduce the left side (matrix A) to the identity matrix I.

  3. To achieve this, you can perform operations such as swapping rows, multiplying a row by a nonzero constant, or adding a multiple of one row to another row.

  4. Apply these row operations to get a 1 in the upper-left corner of matrix A. If the element in the upper-left corner is not 1, divide the entire row by that element to make it 1.

  5. Use row operations to get 0's in all other entries in the first column, except for the element in the second row. To achieve this, multiply the first row by a suitable constant and add it to the second row.

  6. Next, make the element in the second row and second column (the pivot element) equal to 1. If it is not 1, divide the entire second row by that element.

  7. Use row operations to get 0's in all other entries in the second column, except for the element in the first and third rows. Again, multiply the second row by a suitable constant and add it to the first and third rows.

  8. Continue this process, making diagonal elements 1 and reducing all other entries in their respective columns to 0. Repeat steps 4 to 7 for the third row and third column.

  9. Once you have transformed the matrix A into the identity matrix I, the matrix on the right side will be the inverse matrix B. Write B as [I|B].

  10. The matrix [I|B] can be split back into two separate matrices: B is the inverse matrix of A.

  Note: It's important to check if the given matrix is invertible before attempting to find its inverse. A matrix is invertible (has an inverse) if its determinant is nonzero.