Question

By using properties of determinants ,show that: $\begin{vmatrix}0&a&-b\\\\-a&0&-c\\\b&c&0\end{vmatrix}$=0

Answer 1

We have,
△= $\begin{vmatrix}0&a&-b\\\\-a&0&-c\\\b&c&0\end{vmatrix}$
Applying R1 $\to$ cR1,we have

△=$\frac{1}{c}\begin{vmatrix}0&ac&-bc\\\\-a&0&-c\\\b&c&0\end{vmatrix}$
Applying R11$\to$ R1-bR2,we have

△=$\frac{1}{c}\begin{vmatrix}ab&ac&0\\\\-a&0&-c\\\b&c&0\end{vmatrix}$

=$\frac{1}{c}\begin{vmatrix}b&c&0\\\\-a&0&-c\\\b&c&0\end{vmatrix}$

Here, the two rows R1 and R3 are identical.
∴∆ = 0.