My teacher told that in a determinant if we do some operations on a row or column then the coefficient of that should be 1 otherwise the whole determinant should be multiplied with that quantity. I am confused over his sentence... If we take for instance if C1 goes to xC1 - C2 then column 1 should only be multiplied with "x" Right?... How come the whole determinant is getting multiplied with "x"???
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What is CC2? Did you mean just C2? – Hume2 Feb 11 '21 at 10:25
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@Hume2 yes only C2... Sorry – Faham J Feb 11 '21 at 10:31
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2The determinant is a $n-$linear form w.r.t. the columns of the matrix. Hence if you multiply one column by a coefficient $\lambda$, the whole determinant is multiplied by $\lambda$. – TheSilverDoe Feb 11 '21 at 10:33
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You may know that $\det(XY) = \det(X)\det(Y)$. You can represent the row and column operations by multiplying the matrix by another matrix from left (for row operations) or from right (for column operations). If you multiply your matrix by another matrix with determinant equal to one, the product has the same determinant as the original matrix.
The operation which you ask about can be represented by multiplying a matrix from right by this matrix:$$ Y=\begin{bmatrix} x&0&0&0\\ -1&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix} $$ And you see that $\det(Y)=x$, not 1. Therefore it scales the determinant by factor $x$.
If your operation was C1 going to C1-xC2, the matrix would be:$$ Y=\begin{bmatrix} 1&0&0&0\\ -x&1&0&0\\ 0&0&1&0\\ 0&0&0&1 \end{bmatrix} $$ In this case, $\det(Y)=1$, so it doesn't change the determinant.
Hume2
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