Question

The product of all the factors of determinant

$\left|\begin{array}{ccc}x& y& 1\\ x^2& y^2& 1\\ x^3& y^3& 1\end{array}\right|$ is:

• A. xy(x-1)(y-1)(y-x)
• B. xy(x-1)
• C. xy(x+1)(y-x)
• D. (x+1)(x-1)

Answer 1

The correct option is A

xy(x-1)(y-1)(y-x)

Observe carefully, what are the possible ways of making the determinant 0.

If we put x=0 then the first column becomes 0 and hence the determinant becomes 0.

So x-0 is one factor.

Similarly y-0 is the factor.

Now putting x=1 makes first and third columns identical. And we know that when two rows or columns are identical the determinant is 0

So x-1 is the factor.

Similarly y-1 is the factor as putting y=1 makes second and third columns identical.

Also if x=y then first and second columns becomes identical. So y-x is a factor

So product of all the factors is given by option (a)