Certainly! Let's consider a 3x3 matrix as an example: 

    | a  b  c |
M = | d  e  f |
    | g  h  i |

To find the determinant of this matrix using the Laplace expansion method, we can choose any row or column and expand it using cofactors. Let's expand along the first column for this example.

The Laplace expansion formula for the determinant of a 3x3 matrix is:

det(M) = a(det(M11)) - d(det(M12)) + g(det(M13))

Where M11, M12, and M13 represent the sub-matrices obtained by removing the first column and the row of each corresponding element.

M11 = | e  f |     M12 = | d  f |     M13 = | d  e |
      | h  i |           | g  i |           | g  h |

Using this formula, we can calculate the determinant step by step:

det(M) = a(det(M11)) - d(det(M12)) + g(det(M13))
       = a(ei - fh) - d(di - fg) + g(dh - eg)
       = aei - afh - ddi + dfg + gdh - geg

This is the final result, which represents the determinant of the original matrix M using the Laplace expansion method.

It's important to note that this method can be time-consuming for larger matrices, as it requires computing determinants of numerous sub-matrices.
Certainly! Let's walk through an example of finding the determinant of a 3x3 matrix using the Laplace expansion method:

Consider the matrix A:
A = [1 2 3]
     [4 5 6]
     [7 8 9]

Step 1: Choose a row or column to expand along. Let's choose the first column.

Step 2: Calculate the cofactor of each entry in the chosen row or column. The cofactor of an entry is the determinant of the submatrix formed by removing the row and column containing that entry.

For the first column:
C11 = -1^(1+1) * det([5 6] [8 9]) = -1 * (5*9 - 6*8) = -3
C21 = -1^(2+1) * det([2 3] [8 9]) = 1 * (2*9 - 3*8) = -6
C31 = -1^(3+1) * det([2 3] [5 6]) = -1 * (2*6 - 3*5) = 3

Step 3: Multiply each cofactor by its corresponding entry in the chosen row or column.

For the first column:
|A| = 1 * C11 + 4 * C21 + 7 * C31
    = 1 * (-3) + 4 * (-6) + 7 * 3
    = -3 - 24 + 21
    = -6

So, the determinant of matrix A using the Laplace expansion method is -6.