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3 and 4 .Determinants and Matrices
medium
નિશ્ચાયકના ગુણધર્મનો ઉપયોગ કરીને સાબિત કરો : $\left|\begin{array}{ccc}0 & a & -b \\ -a & 0 & -c \\ b & c & 0\end{array}\right|=0$
Option A
Option B
Option C
Option D
Solution
We have,
$\Delta=\left|\begin{array}{ccc}0 & a & -b \\ -a & 0 & -c \\ b & c & 0\end{array}\right|$
Applying $R_{1} \rightarrow c R_{1},$ we have:
$\Delta=\frac{1}{c}\left|\begin{array}{ccc}0 & a c & -b c \\ -a & 0 & -c \\ b & c & 0\end{array}\right|$
Applying $R_{1} \rightarrow R_{1}-b R_{2},$ we have:
$\Delta=\frac{1}{c}\left|\begin{array}{ccc}a b & a c & 0 \\ -a & 0 & -c \\ b & c & 0\end{array}\right|$
$=\frac{a}{c}\left|\begin{array}{ccc}b & c & 0 \\ -a & 0 & -c \\ b & c & 0\end{array}\right|$
Here, the two rows $R_{1}$ and $R_{3}$ are identical.
$\therefore \Delta=0$
Standard 12
Mathematics
