Show that If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.

Answer

To show that the product of two matrices can be a zero matrix without either of the matrices being a zero matrix, we can use a specific example. Let's follow these steps:

### Step 1: Define the Matrices
Let’s define two matrices \( A \) and \( B \) as follows:

\[ A = \begin{pmatrix} 2 & -1 \\ -2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \]

### Step 2: Check if the Matrices are Non-Zero
We need to confirm that both matrices \( A \) and \( B \) are not zero matrices.

- Matrix \( A \) has elements \( 2, -1, -2, 1 \), which are not all zero.
- Matrix \( B \) has elements \( 1, 2, 2, 4 \), which are also not all zero.

### Step 3: Multiply the Matrices
Now, we will calculate the product \( AB \).

\[ AB = \begin{pmatrix} 2 & -1 \\ -2 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \]

To find the product, we will use the row-by-column multiplication method:

1. First row, first column:
\[ 2 \times 1 + (-1) \times 2 = 2 - 2 = 0 \]

2. First row, second column:
\[ 2 \times 2 + (-1) \times 4 = 4 - 4 = 0 \]

3. Second row, first column:
\[ -2 \times 1 + 1 \times 2 = -2 + 2 = 0 \]

4. Second row, second column:
\[ -2 \times 2 + 1 \times 4 = -4 + 4 = 0 \]

Putting it all together, we get:

\[ AB = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 4: Conclusion
The product \( AB \) is the zero matrix, even though neither \( A \) nor \( B \) is a zero matrix. This demonstrates that the product of two matrices can be a zero matrix without either matrix being a zero matrix.