If a, b, c are non-zero real numbers and the | vedclass

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

Question

(B) The given system of equations can be written as:$(a - 1)x - y - z = 0$$-x + (b - 1)y - z = 0$$-x - y + (c - 1)z = 0$For a non-trivial solution,the

Vedclass · Accepted Answer

$abc$

Vedclass · Answer

(B) The given system of equations can be written as:$(a - 1)x - y - z = 0$$-x + (b - 1)y - z = 0$$-x - y + (c - 1)z = 0$For a non-trivial solution,the determinant of the coefficient matrix must be zero:$\begin{vmatrix} a - 1 & -1 & -1 \ -1 & b - 1 & -1 \ -1 & -1 & c - 1 \end{vmatrix} = 0$Applying row operations $R_2 	o R_2 - R_3$ and $R_3 	o R_3 - R_1$:$\begin{vmatrix} a - 1 & -1 & -1 \ 0 & b & -c \ -a & 0 & c \end{vmatrix} = 0$Expanding along the first row:$(a - 1)(bc - 0) + 1(0 - ac) - 1(0 + ab) = 0$$(a - 1)(bc) - ac - ab = 0$$abc - bc - ac - ab = 0$$abc = ab + bc + ca$Thus,$ab + bc + ca = abc$.

If $a, b, c$ are non-zero real numbers and the system of equations $(a - 1)x = y + z,$ $(b - 1)y = z + x,$ $(c - 1)z = x + y$ has a non-trivial solution,then $ab + bc + ca$ equals

Similar Questions

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

If the system of homogeneous equations $\begin{aligned} & t x+(t+1) y+(t-1) z=0 \\ & (t+1) x+t y+(t+2) z=0 \\ & (t-1) x+(t+2) y+t z=0\end{aligned}$ in $x, y, z$ has a non-trivial solution, then $t$ is a root of the equation

If $f(x) = \begin{vmatrix} 1 & x & x+1 \\ 2x & x(x-1) & (x+1)x \\ 3x(x-1) & x(x-1)(x-2) & (x+1)x(x-1) \end{vmatrix}$, then $f(100)$ is equal to:

If $x^4+y^4+z^4=0$ then,$\left|\begin{array}{ccc}1 & xy & yz \\ zx & 1 & xy \\ yz & zx & 1\end{array}\right|=$ . . . . . . . $(\because x, y, z \in \mathbb{R})$

The value of $\left| {\begin{array}{ccc} {1^2} & {2^2} & {3^2} \\ {2^2} & {3^2} & {4^2} \\ {3^2} & {4^2} & {5^2} \end{array}} \right|$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module