Q1Multiple correct5 Marks15 Mar 2026

Let $$A=\begin{bmatrix}1&a&a^2&a^3\\1&b&b^2&b^3\\1&c&c^2&c^3\\1&d&d^2&d^3\end{bmatrix}$$ where $a,b,c,d$ are distinct real numbers. Which of the following statements are correct?

A

If the sum of every row of $A$ is $0$, then $\det(A)=0$.

B

$\det(A)$ changes sign if rows $2$ and $4$ are interchanged.

C

$\det(A)=0$ if any two of the numbers $a,b,c,d$ are equal.

D

$\det(A)$ changes sign if each row is multiplied by $-1$.