If A is a square matrix of order n such that its elements are polynomi...
Explanation:
Let A be a square matrix of order n such that its elements are polynomial in x. Let the r-rows of A become identical for x = k. Then we can write:
A = [a1(x), a2(x), ..., ar-1(x), ar(x), ..., an(x)]
where ar(x) = ar-1(x) for x = k. Let B be the matrix obtained from A by subtracting ar-1(x) from ar(x) in the r-th row. Then we have:
B = [a1(x), a2(x), ..., ar-1(x), 0, ..., an(x)] - [0, 0, ..., 0, ar-1(x) - ar-1(x), ..., 0]
= [a1(x), a2(x), ..., ar-1(x), 0, ..., an(x)] - [0, 0, ..., 0, 0, ..., 0]
= [a1(x), a2(x), ..., ar-1(x), ar-1(x) - ar(x), ..., an(x)]
= [a1(x), a2(x), ..., ar-1(x), (x - k)ar-1(x), ..., an(x)]
Therefore, the r-th row of B is:
[(x - k)ar-1(x), 0, ..., 0]
Now let C be the matrix obtained from B by dividing the r-th row by (x - k)r. Then we have:
C = [a1(x), a2(x), ..., ar-1(x), ar-1(x) / (x - k)r, ..., an(x)]
Therefore, the determinant of C is:
|C| = (ar-1(x) / (x - k)r) times the determinant of the matrix obtained from C by deleting the r-th row.
Since ar-1(x) = ar(x) for x = k, we have:
|C| = (ar-1(k) / (k - k)r) times the determinant of the matrix obtained from C by deleting the r-th row.
= 0 times the determinant of the matrix obtained from C by deleting the r-th row.
Therefore, (x - k)r is a factor of |C|. Since A is obtained from C by multiplying the r-th row by (x - k)r, we have:
|A| = (x - k)r times |C|
Therefore, (x - k)r is a factor of |A|. Hence, option (A) is correct.