If left| {begin{array}{*{20}{c}}{ - {a^2}}&{a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given determinant is $\Delta = \left| {\begin{array}{*{20}{c}}{ - {a^2}}&{ab}&{ac}\{ab}&{ - {b^2}}&{bc}\{ac}&{bc}&{ - {c^2}}\end{array}} ight|

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(C) Given determinant is $\Delta = \left| {\begin{array}{*{20}{c}}{ - {a^2}}&{ab}&{ac}\{ab}&{ - {b^2}}&{bc}\{ac}&{bc}&{ - {c^2}}\end{array}} ight|$.Taking $a, b, c$ common from $R_1, R_2, R_3$ respectively:$\Delta = (abc) \left| {\begin{array}{*{20}{c}}{ - a}&b&c\a&{ - b}&c\a&b&{ - c}\end{array}} ight|$.Taking $a, b, c$ common from $C_1, C_2, C_3$ respectively:$\Delta = (abc)(abc) \left| {\begin{array}{*{20}{c}}{ - 1}&1&1\1&{ - 1}&1\1&1&{ - 1}\end{array}} ight| = {a^2}{b^2}{c^2} \left| {\begin{array}{*{20}{c}}{ - 1}&1&1\1&{ - 1}&1\1&1&{ - 1}\end{array}} ight|$.Now,expanding the determinant:$\left| {\begin{array}{*{20}{c}}{ - 1}&1&1\1&{ - 1}&1\1&1&{ - 1}\end{array}} ight| = -1(1 - 1) - 1(-1 - 1) + 1(1 - (-1)) = 0 + 2 + 2 = 4$.Therefore,$\Delta = 4{a^2}{b^2}{c^2}$.Comparing with $K{a^2}{b^2}{c^2}$,we get $K = 4$.

If $\left| {\begin{array}{*{20}{c}}{ - {a^2}}&{ab}&{ac}\\{ab}&{ - {b^2}}&{bc}\\{ac}&{bc}&{ - {c^2}}\end{array}} \right| = K{a^2}{b^2}{c^2},$ then $K = $

Similar Questions

Evaluate the determinant: $\left|\begin{array}{rrr}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$

If $A = \begin{bmatrix} 1 & 0 & 1 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{bmatrix}$, then $\det(A)$ is equal to

If $abc \neq 0$ and the system of equations $x+7ay+2az=0$,$x+6by+2bz=0$,$x+5cy+2cz=0$ has a non-trivial solution,then $a, b, c$ are in

If the points with position vectors $60 \hat{i}+3 \hat{j}$,$40 \hat{i}-8 \hat{j}$,and $a \hat{i}-52 \hat{j}$ are collinear,then $a$ is equal to

The value of the determinant $\left| \begin{array}{ccc} 4 & -6 & 1 \\ -1 & -1 & 1 \\ -4 & 11 & -1 \end{array} \right|$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module