1.6k views

Which one of the following does NOT equal

$$\begin{vmatrix} 1 & x & x^{2}\\ 1& y & y^{2}\\ 1 & z & z^{2} \end{vmatrix} \quad ?$$

1. $$\begin{vmatrix} 1& x(x+1) & x+1\\ 1& y(y+1) & y+1\\ 1& z(z+1) & z+1 \end{vmatrix}$$
2. $$\begin{vmatrix} 1& x+1 & x^{2}+1\\ 1& y+1 & y^{2}+1\\ 1& z+1 & z^{2}+1 \end{vmatrix}$$
3. $$\begin{vmatrix} 0& x-y & x^{2}-y^{2}\\ 0 & y-z & y^{2}-z^{2}\\ 1 & z & z^{2} \end{vmatrix}$$
4. $$\begin{vmatrix} 2& x+y & x^{2}+y^{2}\\ 2 & y+z & y^{2}+z^{2}\\ 1 & z & z^{2} \end{vmatrix}$$
edited | 1.6k views
+2
This type of question I'm Go with options

a=1,b=2,c=3

and easily get option (A)

Operations are:

$C_{3} \leftarrow C_{3} + C_{2}$

$C_{2} \leftarrow C_{2} + C_{1}$

Swap $C_{2}\, \& \,C_{3}$

The Swapping operations mske the determinant as $(-1)*|A|$

where as the other options have their determinant as $|A|.$

edited
0
why i can not use exchange command here?
+3
Bcz ,determinant property-- if any  two rows (or any two columns)of a determinant are interchanged then the value of determinant is multiplied by "-1".
+7

This is special type of matrix called vandermonde matrix, it is special as it has some important applications.

$\begin{vmatrix} 1 &x &x^{2} \\ 1 & y& y^{2}\\ 1& z& z^{2} \end{vmatrix}$

$(B):$    $C2\rightarrow C2+C1 ,\ C3\rightarrow C3+C1$

$\begin{vmatrix} 1 &x+1 &x^{2}+1 \\ 1 & y+1& y^{2}+1\\ 1& z+1& z^{2}+1 \end{vmatrix}$

$(C):$   $R1\rightarrow R1-R2 ,\ R2\rightarrow R3-R2$

$\begin{vmatrix} 0 &x-y &x^{2}-y^{2} \\ 0 & y-z& y^{2}-z^{2}\\ 1& z& z^{2} \end{vmatrix}$

$(D):$   $R1\rightarrow R1+R2 ,\ R2\rightarrow R3+R2$

$\begin{vmatrix} 2 &x+y &x^{2}+y^{2} \\ 2 & y+z& y^{2}+z^{2}\\ 1& z& z^{2} \end{vmatrix}$

We can't get option(A) from given Determinant.

Hence,Option(A) is the correct choice.
edited
the correct ans is option (A)

1
2