Question

If the matrix ${\displaystyle \left[\begin{array}{ccc}0& \text{a}& 3\\ 2& \text{b}& -1\\ \text{c}& 1& 0\end{array}\right]}$, is a skew symmetric matrix, find the values of a, b and c.

Answer 1

Let A = ${\displaystyle \left[\begin{array}{ccc}0& \text{a}& 3\\ 2& \text{b}& -1\\ \text{c}& 1& 0\end{array}\right]}$

A = ${\displaystyle \left[\begin{array}{ccc}0& 2& \text{c}\\ \text{a}& \text{b}& 1\\ 3& -1& 0\end{array}\right]}$

For skew symmetric matrix, A' = – A.

⇒ ${\displaystyle \left[\begin{array}{ccc}0& 2& \text{c}\\ \text{a}& \text{b}& 1\\ 3& -1& 0\end{array}\right]=-\left[\begin{array}{ccc}0& \text{a}& 3\\ 2& \text{b}& -1\\ \text{c}& 1& 0\end{array}\right]}$

⇒ ${\displaystyle \left[\begin{array}{ccc}0& 2& \text{c}\\ \text{a}& \text{b}& 1\\ 3& -1& 0\end{array}\right]=\left[\begin{array}{ccc}0& -\text{a}& -3\\ -2& -\text{b}& 1\\ -\text{c}& -1& 0\end{array}\right]}$

Equating the corresponding elements, we get

a = – 2, b = – b

⇒ 2b = 0

⇒ b = 0

And c = – 3

Hence, a = – 2, b = 0 and c = – 3.