If, we have any given matrix A and we multiply any row or column of matrix with constant k then determinant of modified matrix becomes k times of determinant of A. We can show this by taking example of 3 x 3 matrix A and calculating its determinant.
Lets take an example of 3 x 3 matrix
|A|= a(cofactor of
)+b(cofactor of
)+c(cofactor of
)
(1)
Now, I multiply one row of matrix A with k and I get matrix
|B|= a(cofactor of
)+b(cofactor of
)+c(cofactor of
)
(2)
From equation (1) and (2), I can say that k|A|=|B|
Therefore, it is showed that determinant of matrix becomes k times if we multiply any of its row by k. Same thing can be proved if, we multiply any column by k. Note that, if we multiply any two rows of matrix by k then value of its determinant becomes
times. If, we multiply every row of matrix of order n x n by k then value of its determinant becomes
times.
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