# What happens to the determinant of a matrix when a row of the matrix is multiplied by a constant?

What will happen to the following matrix if the third row is multiplied by 3?

3 3 3
1 5 6
3 4 5


Also, in general, how does scaling a row of a square matrix affect its determinant?

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## 2 Answers

The determinant is multiplied by the scaling factor. You can see this from the definition of the determinant as the signed sum of all products with one factor from each row and column – since each summand contains exactly one factor from the scaled row, each summand is scaled by the scaling factor, and thus so is the sum.

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What will happen if a column is multiplied by a constant? Will it have the same effect? –  Anderson Green Sep 26 '12 at 20:31
@Anderson: Since "row" and "column" play equivalent roles in the definition of the determinant as the signed sum of all products with one factor from each row and column, all properties of determinants that hold for rows also hold for columns and vice versa. –  joriki Sep 26 '12 at 20:33
Does this mean that the determinant of the transpose of a matrix is the same as the determinant of the original matrix? –  Anderson Green Sep 26 '12 at 20:43
@Anderson: Indeed it does. –  joriki Sep 26 '12 at 20:48

If you multiply a row by $n$, the determinant is multiplied by $n$.

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