Question

If A and B are skew symmetric matrices then (AB - BA) is a matrix and (AB + BA) is a matrix.

• A. skew - symmetric
• B. symmetric matrix
• C. null matrix
• D. identity matrix

Answer 1

Answer: (A), (B)

The correct options are
A skew - symmetric
B symmetric matrix

$A\mathrm{and}B\mathrm{are}\mathrm{skew}-\mathrm{symmetric}\mathrm{matrices}\Rightarrow A^T=-A,B^T=-B\mathrm{Now},{\left(\mathrm{AB}+\mathrm{BA}\right)}^T=B^T{A}^T+A^T{B}^T=\left(-B\right)\left(-A\right)+\left(-A\right)\left(-B\right)=\mathrm{BA}+\mathrm{AB}\Rightarrow (\mathrm{AB}+\mathrm{BA})\mathrm{is}\mathrm{symmetric}{\left(\mathrm{AB}-\mathrm{BA}\right)}^T=B^T{A}^T-A^T{B}^T=\left(-B\right)\left(-A\right)-\left(-A\right)\left(-B\right)=\mathrm{BA}-\mathrm{AB}=-(\mathrm{AB}-\mathrm{BA})\Rightarrow (\mathrm{AB}-\mathrm{BA})\mathrm{is}\mathrm{skew}-\mathrm{symmetric}$