Jayden Davidson

2022-12-18

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

siabrukbax

Expert

One way to think about it, using the property: $det\left(AB\right)=det\left(A\right)det\left(B\right)$:
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:
$\left[\begin{array}{cccccc}1& & & & & \\ & \ddots & & & & & \\ & & 1& & & & \\ & & & \ddots & & & \\ & & k& & 1& & \\ & & & & & \ddots & \\ & & & & & & 1\end{array}\right]$
where ${a}_{ji}=k$
The determinant of a triangular matrix is the product of the diagonal. A has a unit diagonal, so det(A)=1.
Therefore,
$det\left(AB\right)=det\left(A\right)det\left(B\right)=1det\left(B\right)=det\left(B\right).$

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