$\left|\begin{array}{lll}1 & 3 & 6 \\ 6 & 1 & 4 \\ 3 & 7 & 12\end{array}\right|+4\left|\begin{array}{lll}2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6\end{array}\right|=10\left|\begin{array}{lll}1 & 2 & 1 \\ 3 & 1 & 7 \\ 3 & 2 & 6\end{array}\right|$
In 1st determinant, taking 2 common from $C_3$ we get
L.H.S. $=\left|\begin{array}{ccc}1 & 3 & 6 \\ 6 & 1 & 4 \\ 3 & 7 & 12\end{array}\right|+4\left|\begin{array}{lll}2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6\end{array}\right|$
In $1^{\text {st }}$ determinant, taking 2 common from $\mathrm{C}_3$,
we get
L.H.S. $=2\left|\begin{array}{lll}1 & 3 & 3 \\ 6 & 1 & 2 \\ 3 & 7 & 6\end{array}\right|+4\left|\begin{array}{lll}2 & 3 & 3 \\ 2 & 1 & 2 \\ 1 & 7 & 6\end{array}\right|$
$=\left|\begin{array}{lll}2 & 3 & 3 \\ 12 & 1 & 2 \\ 6 & 7 & 6\end{array}\right|+\left|\begin{array}{lll}8 & 3 & 3 \\ 8 & 1 & 2 \\ 4 & 7 & 6\end{array}\right|$
$=\left|\begin{array}{ccc}2+8 & 3 & 3 \\ 12+8 & 1 & 2 \\ 6+4 & 7 & 6\end{array}\right|$
$=\left|\begin{array}{lll}10 & 3 & 3 \\ 20 & 1 & 2 \\ 10 & 7 & 6\end{array}\right|$
Interchanging rows and columns, we get
L.H.S. $=\left|\begin{array}{ccc}10 & 20 & 10 \\ 3 & 1 & 7 \\ 3 & 2 & 6\end{array}\right|$
Taking 10 common from $R_1$, we get
L. H.S $=10\left|\begin{array}{lll}1 & 2 & 1 \\ 3 & 1 & 7 \\ 3 & 2 & 6\end{array}\right|=$ R.H.S.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| X | $0$ | $1$ | $2$ | $3$ | $4$ |
| P(X) | $0.1$ | $0.5$ | $0.2$ | $− 0.1$ | $0.2$ |