Question

Let the homogeneous system of linear equations $px+y+z=0,x+qy+z=0$, and $x+y+rz=0$, where $p,q,r\ne 1,$ have a non-zero solution, then the value of $\frac{1}{1-p}+\frac{1}{1-q}+\frac{1}{1-r}$ is.

• A. $-1$
• B. $0$
• C. $2$
• D. $1$

Answer 1

The correct option is D $1$

$\left|\begin{array}{ccc}p& 1& 1\\ 1& q& 1\\ 1& 1& r\end{array}\right|=0$

$R_1\to R_1-R_2\&R_2\to R_2-R_3$

$\left|\begin{array}{ccc}p-1& 1-q& 0\\ ⊝p& q-1& 1⊝r\\ 1& 1& r\end{array}\right|=0$

$p-1\left(\right(q-1)+(r-1\left)\right)+(q-1)(r-1)=0$ dividing by $(1-p)(1-q)(1-r)=0$

$\frac{1}{1-p}+\frac{1}{1-q}+\frac{1}{1-r}=1$

Option D