To find the determinant of a square matrix, you need to perform certain operations based on its size. Here are the methods for calculating the determinant for different sizes of matrices:

  1. 2x2 Matrix:

  For a 2x2 matrix

   [a b]

   [c d],

  the determinant is given by the formula: ad - bc.

  2. 3x3 Matrix:

  For a 3x3 matrix

   [a b c]

   [d e f]

   [g h i],

  one way to calculate the determinant is by using the formula known as Sarrus' Rule. The determinant can be calculated as follows:

   det = aei + bfg + cdh - ceg - afh - bdi.

  Another method for calculating the determinant of a 3x3 matrix is by using the Laplace Expansion. You can select any row or column and calculate the determinant by expanding along that row or column. For example, if you expand along the first row, the formula would be:

   det = a*cofactor(A11) - b*cofactor(A12) + c*cofactor(A13),

   where cofactor(Aij) is the determinant of the 2x2 matrix obtained by deleting the row and column containing element Aij.

  3. NxN Matrix:

  For an N x N matrix, where N is greater than 3, you can use a similar approach called the Laplace Expansion. Expand along any row or column and calculate the determinant using cofactors of the minor matrices.

  It is important to note that the determinant of a matrix has various properties, such as linearity in rows or columns, that can be used to simplify calculations. Additionally, there exist other methods, such as using row operations to convert the matrix to an upper or lower triangular form, which simplifies the determinant calculation.

  Determinants have numerous applications in various fields such as linear algebra, calculus, physics, and engineering.