Question

One of the factor of $\left|\begin{array}{ccc}{p}^2+x& pq& pr\\ pq& q^2+x& qr\\ pr& qr& r^2+x\end{array}\right|$ is

• A. $x$
• B. $(p^2+x)(q^2+x)(r^2+x)$
• C. $(p^2+q^2+r^2+x)$
• D. None of these

Answer 1

The correct option is C $(p^2+q^2+r^2+x)$

$\left|\begin{array}{ccc}{p}^2+x& pq& pr\\ pq& q^2+x& qr\\ pr& qr& r^2+x\end{array}\right|$
$\frac{1}{pqr}\left|\begin{array}{ccc}p(p^2+x)& p^{2}q& p^{2}r\\ p{q}^2& q(q^2+x)& q^{2}r\\ p{r}^2& q{r}^2& r(r^2+x)\end{array}\right|$
$\left|\begin{array}{ccc}(p^2+x)& p^2& p^2\\ q^2& (q^2+x)& q^2\\ r^2& r^2& (r^2+x)\end{array}\right|$
$R_1\to R_1+R_2+R_3$
$\left|\begin{array}{ccc}(p^2+q^2+r^2+x)& (p^2+q^2+r^2+x)& (p^2+q^2+r^2+x)\\ q^2& (q^2+x)& q^2\\ r^2& r^2& (r^2+x)\end{array}\right|$
$(p^2+q^2+r^2+x)\left|\begin{array}{ccc}1& 1& 1\\ q^2& (q^2+x)& q^2\\ r^2& r^2& (r^2+x)\end{array}\right|$
$C_1\to C_1-C_3$
$(p^2+q^2+r^2+x)\left|\begin{array}{ccc}0& 1& 1\\ 0& (q^2+x)& q^2\\ -x& r^2& (r^2+x)\end{array}\right|$
Clearly one factor of the determinant is $(p^2+q^2+r^2+x)$