Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Laplace expansion along a column

Builds on DeterminantCofactor

Statement

Let KK be a commutative ring, let n≥2n \ge 2, and let A=(aij)A = (a_{ij}) be an n×nn \times n matrix over KK. Let CijC_{ij} be the cofactors of AA. Then, for each column jj, det⁡A=∑i=1naijCij=∑i=1n(−1)i+jaijMij.\det A = \sum_{i=1}^{n} a_{ij} C_{ij} = \sum_{i=1}^{n} (-1)^{i+j} a_{ij} M_{ij}.

Remarks

It follows from the expansion along a row, applied to ATA^T, because det⁡AT=det⁡A\det A^T = \det A. For example, expansion along column 11 of (123045067)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 6 & 7 \end{pmatrix} gives 1⋅(28−30)=−21 \cdot (28 - 30) = -2.

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  9. Minor definition
  10. Cofactor definition
  11. Laplace expansion along a column statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Laplace_expansion, lead section
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Kindstatement, theorem
WritingW0 stub
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