Question

Prove that
$\left|\begin{array}{ccc}p+a& p+b& p+c\\ q+a& q+b& q+c\\ r+a& r+b& r+c\end{array}\right|=0$

Answer 1

$\left|\begin{array}{ccc}p+a& p+b& p+c\\ q+a& q+b& q+c\\ r+a& r+b& r+c\end{array}\right|=0$

L.H.S.

$\left|\begin{array}{ccc}p+a& p+b& p+c\\ q+a& q+b& q+c\\ r+a& r+b& r+c\end{array}\right|$

$R_1\to R_1-R_2$

$R_2\to R_2-R_3$

$\left|\begin{array}{ccc}p-q& p-q& p-q\\ q-r& q-r& q-r\\ r+a& r+b& r+c\end{array}\right|$

$(p-q)(q-r)\left|\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ r+a& r+b& r+c\end{array}\right|$

We know that,

When two rows or two columns are equal in a matrix or determinant the value of determinant is zero.

So, $(p-q)(q-r)\times 0=0$.