Can someone explain why a row replacement operation does not change the determinant of a matrix?

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siabrukbax · Accepted Answer

One way to think about it, using the property:   det  (  A  B  )  =  det  (  A  )  det  (  B  ):Adding a multiple of one row to another is equivalent to left multiplication by an elementary matrix.Let B be some   n  ×  n matrix, A be an n×n elementary matrix which acts as an operator which adds k copies of row i to row j. So applying that same row operation to B will result in the matrix AB. Then without a loss of generality, A has the form:      [                            1                                                                                      ⋱                                                                                              1                                                                                              ⋱                                                                              k                                  1                                                                                              ⋱                                                                                              1                      ]  where       a          j      i        =  kThe determinant of a triangular matrix is the product of the diagonal. A has a unit diagonal, so det(A)=1.Therefore,  det  (  A  B  )  =  det  (  A  )  det  (  B  )  =  1  det  (  B  )  =  det  (  B  )  .

Can someone explain why a row replacement operation does not change the determinant of a matrix?

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