Question

Check whether (l + m + n) is a factor of the determinant

|(1 + m, m + n, n +1), (n, 1, m), (2, 2, 2)|

or not. Give reason.

Answer 1

Apply R₁ → R₁ + R₂ 

|(1 + m + n, m + n + 1, n + 1 + m), (n, 1, m), (2, 2, 2)|

= 2(1 + m + n)|(1, 1, 1), (n, 1, m), (1, 1, 1)|;

yes(1 + m + n) is a factor