Let l, m, n in R and A = begin{bmatrix} 1 & r | vedclass

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(C) The rank of matrix $A$ is $3$ if and only if there exists at least one $3 	imes 3$ minor with a non-zero determinant. Consider the submatrix $M$

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$R - \{1\}$

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(C) The rank of matrix $A$ is $3$ if and only if there exists at least one $3 	imes 3$ minor with a non-zero determinant. Consider the submatrix $M$ formed by the first three columns: $M = \begin{bmatrix} 1 & r & r^2 \ r & r^2 & 1 \ r^2 & 1 & r \end{bmatrix}$. The determinant of $M$ is given by $|M| = 1(r^3 - 1) - r(r^2 - r^2) + r^2(r - r^4) = r^3 - 1 + r^3 - r^6 = -(r^6 - 2r^3 + 1) = -(r^3 - 1)^2$. For the rank to be $3$,we require $|M| 
eq 0$. $-(r^3 - 1)^2 
eq 0 \Rightarrow r^3 - 1 
eq 0 \Rightarrow r^3 
eq 1 \Rightarrow r 
eq 1$. Thus,the rank of $A$ is $3$ for all $r \in R - \{1\}$.

Let $l, m, n \in R$ and $A = \begin{bmatrix} 1 & r & r^2 & l \\ r & r^2 & 1 & m \\ r^2 & 1 & r & n \end{bmatrix}$. Then the set of all real values of $r$ for which the rank of $A$ is $3$,is

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