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Laplace Expansion 📂Matrix Algebra

Laplace Expansion

정리

A square matrix An×n=(aij)A_{n \times n} = (a_{ij}) is given.

  • [1]: For the selected ii row det⁡A=∑j=1naijCij \det A = \sum_{j=1}^{n} a_{ij} C_{ij}
  • [2]: For the selected jj column det⁡A=∑i=1naijCij \det A = \sum_{i=1}^{n} a_{ij} C_{ij}
  • The determinant MijM_{ij} of the matrix obtained by removing the iith row and jjth column from the square matrix An×n=(aij)A_{n \times n} = (a_{ij}) is called a minor, and Cij:=(−1)i+jMijC_{ij} := (-1)^{i + j} M_{ij} is called a cofactor.

설명

Laplace expansion is also known as cofactor expansion, and its usefulness is beyond words. It is much easier than calculating the determinant with just the definition. When finding the determinant of a matrix under specific conditions, its advantages increase even more, so it is absolutely essential to know this fact.

예시

Let’s look at the following Laplace expansion as an example that can be used to determine whether a matrix is invertible or not.

det⁡[123456789]=1⋅∣5689∣−2⋅∣4679∣+3⋅∣4578∣=1⋅(−3)−2⋅(−6)+3⋅(−3)=0 \begin{align*} \displaystyle \det \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} =& 1 \cdot \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} - 2 \cdot \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} + 3 \cdot \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix} \\ =& 1 \cdot (-3) - 2 \cdot (-6) + 3 \cdot (-3) \\ =& 0 \end{align*}

Therefore, it is easy to confirm that det⁡[123456789]\displaystyle \det \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} does not have an inverse.

Code

The following is the code for implementing and verifying the Laplace expansion in Julia. It is practically a direct translation of the equations, and in reality, it is implemented very inefficiently. For this, it would be good to refer to the following post: