Show That \(AB = BA = O_{3\times3}\)

Solution:

Key Observation

Matrix \(A\) is skew-symmetric and matrix \(B\) is formed from vector:

\[ \vec{v} = (a, b, c) \] \[ B = \vec{v}^T \vec{v} \]

Step 1: Multiply \(A\) with vector \((a,b,c)\)

\[ A \begin{bmatrix}a\\b\\c\end{bmatrix} = \begin{bmatrix} 0\cdot a + c b – b c \\ -c a + 0\cdot b + a c \\ b a – a b + 0\cdot c \end{bmatrix} = \begin{bmatrix}0\\0\\0\end{bmatrix} \]

So, \(A\vec{v} = 0\)

Step 2: Use structure of \(B\)

\[ B = \vec{v}^T \vec{v} \] Then, \[ AB = A(\vec{v}^T \vec{v}) = (A\vec{v})\vec{v}^T = 0 \]

Step 3: Similarly

\[ BA = \vec{v}^T (\vec{v}A) = 0 \]

Conclusion:

\[ AB = BA = O_{3\times3} \]

Hence proved.