Why Does Sum of Matrix Elements x Cofactor in Different Rows Equal Zero?

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SUMMARY

The discussion centers on the mathematical property that the sum of the elements of a matrix multiplied by the cofactors from different rows equals zero. Specifically, when considering a square matrix A and replacing row j with row i to form a new matrix B, the determinant calculated using elements from row i and cofactors from row j yields the same result as calculating the determinant of matrix B along row j. This demonstrates that the determinant remains unchanged, reinforcing the principle that the sum of these products across different rows results in zero.

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  • Understanding of matrix theory and determinants
  • Familiarity with cofactors and their calculation
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  • Basic linear algebra concepts
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to explain the properties of determinants and cofactors.

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In a matrix the sum of element of a matrix in a row times it's co factor of that elemt gives the determinant value, but why does the sum of element of a matrix times cofactor of different row is always zero?
 
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In a square matrix A, replace row j by row i, obtaining a new matrix B with row i and row j being equal. Expanding using elements from row i in A and cofactors corresponding to row j in A will be the same as expanding det B along row j (using cofactors corresponding to row j in B also) since the elements and cofactors are the same in both cases. So the result is det B, which is...?
 
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