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

Jayden Davidson

Jayden Davidson

Answered

2022-12-18

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

Answer & Explanation

siabrukbax

siabrukbax

Expert

2022-12-19Added 4 answers

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 = k
The 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 ) .

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