Example of Non-Zero Matrices A and B such that AB = O

Example:

\[ A = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} \]

Step: Multiply AB

\[ AB = \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix} = \begin{bmatrix} 1(1) + 1(-1) & 1(-1) + 1(1) \\ 1(1) + 1(-1) & 1(-1) + 1(1) \end{bmatrix} \]

\[ AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O \]

Conclusion:

Both A and B are non-zero matrices, yet their product is the zero matrix.