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

The matrix $\begin{bmatrix} 1 & a & 2 \\ 1 & 2 & 5 \\ 2 & 1 & 1 \end{bmatrix}$ is not invertible if $a$ has the value:

If points $(5, 5)$,$(10, k)$ and $(-5, 1)$ are collinear,then $k =$

If $\left|\begin{array}{lll}a & a^3 & a^4 \\ b & b^3 & b^4 \\ c & c^3 & c^4\end{array}\right|=k(a-b)(b-c)(c-a)$ then $k=$

If $f(x) = \left| \begin{array}{ccc} x-3 & 2x^2-18 & 3x^3-81 \\ x-5 & 2x^2-50 & 4x^3-500 \\ 1 & 2 & 3 \end{array} \right|$,then $f(1)f(3) + f(3)f(5) + f(5)f(1)$ is equal to

The characteristic equation of the matrix $A = \begin{bmatrix} 2 & 3 & 0 \\ 1 & 2 & 5 \\ 3 & -1 & 2 \end{bmatrix}$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module