Question

Let $\Delta \left(x\right)=\left|\begin{array}{ccc}{a}_1+x& b_1+x& c_1+x\\ a_2+x& b_2+x& c_2+x\\ a_3+x& b_3+x& c_3+x\end{array}\right|$ then

• A. $\Delta \left(x\right)$ is a linear polynomial
• B. $\Delta \left(x\right)$ is a polynomial of degree $2$
• C. Coefficient of $x$ in $\Delta \left(x\right)$ is $S$ where $S$ denotes the sum of all the co-factors of the elements in $\Delta \left(0\right)$.
• D. Constant term in $\Delta \left(x\right)$ is $\Delta \left(0\right)$.

Answer 1

Answer: (A), (C), (D)

The correct options are
A $\Delta \left(x\right)$ is a linear polynomial
C Constant term in $\Delta \left(x\right)$ is $\Delta \left(0\right)$.
D Coefficient of $x$ in $\Delta \left(x\right)$ is $S$ where $S$ denotes the sum of all the co-factors of the elements in $\Delta \left(0\right)$.

Given $\Delta \left(x\right)=\left|\begin{array}{ccc}{a}_1+x& b_1+x& c_1+x\\ a_2+x& b_2+x& c_2+x\\ a_3+x& b_3+x& c_3+x\end{array}\right|$
Let $\Delta \left(x\right)=a_0+a_{1}x+a_2{x}^2+....$
$\Rightarrow \Delta \left(x\right)=\left|\begin{array}{ccc}{a}_1+x& b_1+x& c_1+x\\ a_2+x& b_2+x& c_2+x\\ a_3+x& b_3+x& c_3+x\end{array}\right|=a_0+a_{1}x+a_2{x}^2+....$ ....(1)
Put $x=0$ in eqn(1), we get
$\Delta \left(0\right)=a_0=\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$
Differentiating (1) w.r.t. x, we get
$\left|\begin{array}{ccc}1& 1& 1\\ a_2+x& b_2+x& c_2+x\\ a_3+x& b_3+x& c_3+x\end{array}\right|+\left|\begin{array}{ccc}{a}_1+x& b_1+x& c_1+x\\ 1& 1& 1\\ a_3+x& b_3+x& c_3+x\end{array}\right|+\left|\begin{array}{ccc}{a}_1+x& b_1+x& c_1+x\\ a_2+x& b_2+x& c_2+x\\ 1& 1& 1\end{array}\right|=a_1+2a_{2}x+....$ ....(2)
Put $x=0$ in above eqn
$\left|\begin{array}{ccc}1& 1& 1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|+\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ 1& 1& 1\\ a_3& b_3& c_3\end{array}\right|+\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ 1& 1& 1\end{array}\right|=a_1$
$\Rightarrow a_1=(b_2{c}_3-b_3{c}_2)-(a_2{c}_3-a_3{c}_2)+(a_2{b}_3-a_3{b}_2)+(a_1{c}_3-a_1{b}_3)-(b_1{c}_3-b_1{a}_3)+(c_1{b}_3-c_1{a}_3)+(a_1{b}_2-a_1{c}_2)-(a_2{b}_1-b_1{c}_2)+(a_2{c}_1-b_2{c}_1)$
which is the sum of the cofactors of all elements of $\Delta \left(0\right)$
On differentiating eqn (2), we get
$a_2=0$ (As when we differentiate first determinant of eqn (2), we get 3 determinants. In first determinant ,first row being constant term as elements , so will be 0 , while the second determinant and third determinant will have identical rows with elements as 1. So all the three determinants will be 0. Similarly , all the 3 determinants of second matrix will become 0)
Clearly, $\Delta \left(x\right)$ is a linear polynomial.