If, we have a given matrix A. We can perform some column and row operations on this matrix such that value of its determinant remains unchanged. If, we multiply particular row (column) of matrix A with a constant k and then add all the element of that row (column) to the corresponding elements of another row (column) then the value of determinant remains unchanged. This is an important property regarding matrices and determinants.
Lets take an example of 3 x3 matrix
To get matrix B, I have multiplied second row of matrix A with k and added its elements to the corresponding elements of first row of matrix A
According to the property, we should have determinant(A)=determinant(B).
We can verify this by calculating determinant of both A and B.
|A| =a(cofactor of
)+b(cofactor of
) + c(cofactor of
)
|B| = (a+kd)(cofactor of
) + (b+ke) (cofactor of
) + (c+kf) (cofactor of
)
We can see that determinant of A is equal to determinant of B. Similarly, we can prove that determinant of C is equal to determinant of A and B where
To get matrix C, I have multiplied second column of matrix A with k and added its elements to the corresponding elements of column 1 of matrix A.
Leave a Reply