Question

Find the Rank of the matrix
$\left[\begin{array}{cccc}0& c& -b& \alpha \\ -c& 0& a& \beta \\ b& -a& 0& \gamma \\ -\alpha & -\beta & -\gamma & 0\end{array}\right]$
where a, b, c are all positive numbers and $a\alpha +b\beta +c\gamma =0$.

• A. 4
• B. 3
• C. 2
• D. None of these

Answer 1

The correct option is C 2

Given matrix is $\left[\begin{array}{cccc}0& c& -b& \alpha \\ -c& 0& a& \beta \\ b& -a& 0& \gamma \\ -\alpha & -\beta & -\gamma & 0\end{array}\right]$ and $a\alpha +b\beta +c\gamma =0$
The rank of a matrix is said to be r if
1. Every minor of A of order r+1 is zero
2. There is atleast one minor of A of order r which doesnot vanish
given matrix is skew-symmetric matrix
$\to \left|\begin{array}{cccc}0& c& -b& \alpha \\ -c& 0& a& \beta \\ b& -a& 0& \gamma \\ -\alpha & -\beta & -\gamma & 0\end{array}\right|$ = $0$
$\left|\begin{array}{ccc}0& c& -b\\ -c& 0& a\\ b& -a& 0\end{array}\right|=\left|\begin{array}{ccc}0& a& \beta \\ -a& 0& \gamma \\ -\beta & -\gamma & 0\end{array}\right|=0(\because skew-symmetric)\left|\begin{array}{ccc}c& -b& \alpha \\ 0& a& \beta \\ -a& 0& \gamma \end{array}\right|=a\alpha +b\beta +c\gamma =0\left|\begin{array}{ccc}-c& 0& a\\ b& -a& 0\\ -\alpha & -\beta & -\gamma \end{array}\right|=-a(a\alpha +b\beta +c\gamma )=0$
$\left|\begin{array}{cc}0& c\\ -c& 0\end{array}\right|\ne 0$
Hence, rank of the matrix is 2