Show with the help of an example that `AB=O` whereas `BA!=O`, where O is a zero matrix and A,B are both non-zero matrices.

Answer

To demonstrate that \( AB = O \) while \( BA \neq O \) using an example, we will follow these steps:

### Step 1: Define the Matrices
Let us define two non-zero matrices \( A \) and \( B \):
\[ A = \begin{pmatrix} -1 & -1 \\ 1 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix} \]

### Step 2: Calculate \( AB \)
Now, we will calculate the product \( AB \):
\[ AB = \begin{pmatrix} -1 & -1 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix} \]

Calculating the elements of the resulting matrix:
- First row, first column:
\[ (-1 \times 1) + (-1 \times -1) = -1 + 1 = 0 \]
- First row, second column:
\[ (-1 \times 0) + (-1 \times 0) = 0 + 0 = 0 \]
- Second row, first column:
\[ (1 \times 1) + (1 \times -1) = 1 - 1 = 0 \]
- Second row, second column:
\[ (1 \times 0) + (1 \times 0) = 0 + 0 = 0 \]

Thus, we have:
\[ AB = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = O \]

### Step 3: Calculate \( BA \)
Next, we will calculate the product \( BA \):
\[ BA = \begin{pmatrix} 1 & 0 \\ -1 & 0 \end{pmatrix} \begin{pmatrix} -1 & -1 \\ 1 & 1 \end{pmatrix} \]

Calculating the elements of the resulting matrix:
- First row, first column:
\[ (1 \times -1) + (0 \times 1) = -1 + 0 = -1 \]
- First row, second column:
\[ (1 \times -1) + (0 \times 1) = -1 + 0 = -1 \]
- Second row, first column:
\[ (-1 \times -1) + (0 \times 1) = 1 + 0 = 1 \]
- Second row, second column:
\[ (-1 \times -1) + (0 \times 1) = 1 + 0 = 1 \]

Thus, we have:
\[ BA = \begin{pmatrix} -1 & -1 \\ 1 & 1 \end{pmatrix} \neq O \]

### Conclusion
We have shown that:
\[ AB = O \quad \text{and} \quad BA \neq O \]