Determinant of a matrix changes its sign if we interchange any two rows or columns present in a matrix. We can prove this property by taking an example. We take matrix A and we calculate its determinant (|A|).
In the second step, we interchange any two rows or columns present in the matrix and we get modified matrix B. We calculate determinant of matrix B. We will be able to see that determinant of |A|=-|B|.
Now, lets take an example of 3×3 matrix
|A|= a(cofactor of
)+b(cofactor of
)+c(cofactor of
)
(1)
Now lets exchange any two rows or columns and we get matrix
|B|= g(cofactor of
)+h(cofactor of
)+i(cofactor of
)
Therefore, we can see that |A|=-|B| when we exchange any two rows or columns of matrix that is value of determinant changes sign.
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