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

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

Question

(C) The given system of equations is:$(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 c

Vedclass · Accepted Answer

$abc$

Vedclass · Answer

(C) The given system of equations is:$(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$Adding $R_2$ and $R_3$ to $R_1$:$\begin{vmatrix} a-3 & b-3 & c-3 \ -1 & b-1 & -1 \ -1 & -1 & c-1 \end{vmatrix} = 0$Alternatively,expanding the determinant:$(a-1)((b-1)(c-1) - 1) + 1(-(c-1) - 1) - 1(1 + (b-1)) = 0$$(a-1)(bc - b - c + 1 - 1) + (-c + 1 - 1) - (1 + b - 1) = 0$$(a-1)(bc - b - c) - c - b = 0$$abc - ab - ac - bc + b + c - c - b = 0$$abc - ab - ac - bc = 0$Therefore,$ab + bc + ca = abc$.

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

Similar Questions

For how many value$(s)$ of $x$ in the closed interval $[-4, -1]$ is the matrix $\begin{bmatrix} 3 & -1+x & 2 \\ 3 & -1 & x+2 \\ x+3 & -1 & 2 \end{bmatrix}$ singular?

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

The number of real values of $t$ such that the system of homogeneous equations
$\begin{aligned}
t x+(t+1) y+(t-1) z &=0 \\
(t+1) x+t y+(t+2) z &=0 \\
(t-1) x+(t+2) y+t z &=0
\end{aligned}$
has non-trivial solutions is

If $\left|\begin{array}{ll}3 & x \\ x & 1\end{array}\right|=\left|\begin{array}{ll}3 & 2 \\ 4 & 1\end{array}\right|$,then the value of $x$ is . . . . . . .

$A$ value of $\theta$ for which the following system of equations has a non-trivial solution is:
$(4 \sin \theta) x - 3y + z = 0$
$x - (6 \cos 2\theta) y + z = 0$
$3x - 12y + 4z = 0$

Vedclass Test Series

Exam Paper Generator

Online Exam Module