Question

If $A=\left[\begin{array}{ccc}a& b& c\\ x& y& z\\ l& m& n\end{array}\right]$ is a skew-symmetric matrix, then which of the following is equal to x+y+z ?

• A. a+b+c
• B. l+m+n
• C. a-b-m
• D. c-l-n

Answer 1

The correct option is C a-b-m

If $A^T=-A$ then A is called skew-symmetric matrix.

$A=\left[\begin{array}{ccc}a& b& c\\ x& y& z\\ l& m& n\end{array}\right]$

$-A=\left[\begin{array}{ccc}-a& -b& -c\\ -x& -y& -z\\ -l& -m& -n\end{array}\right]$

$A^T=\left[\begin{array}{ccc}a& x& l\\ b& y& m\\ c& z& n\end{array}\right]$

$⟹\left[\begin{array}{ccc}a& x& l\\ b& y& m\\ c& z& n\end{array}\right]=\left[\begin{array}{ccc}-a& -b& -c\\ -x& -y& -z\\ -l& -m& -n\end{array}\right]$

$⟹x=-b,z=-m$

In a skew-symmetric matrix principal diagonal elements should be zero.

Therefore,

$a=y=n=0$

Therefore,

$x+y+z=-b+a-m=a-b-m$