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The question is: 

**True or False? If \( A \) and \( B \) are \( n \times n \) matrices with \( AB = 0 \), then \( A = 0 \) or \( B = 0 \). (\( 0 \) denotes the \( n \times n \) matrix with all entries equal to zero.)**

### Analysis:
The given statement is generally false. Here's why:

For matrices \( A \) and \( B \), the product \( AB = 0 \) does not necessarily imply that either \( A \) or \( B \) must be the zero matrix. This can be shown with a counterexample.

### Counterexample:
Consider the following \( 2 \times 2 \) matrices:
\[ A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]

The product \( AB \) is:
\[ AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \]

Clearly, \( AB = 0 \) but neither \( A \) nor \( B \) is the zero matrix.

Therefore, the correct answer to the question is **False**.

Would you like more details or have any questions?

### Related Questions:
1. What are the conditions under which the product of two matrices is the zero matrix?
2. Can you provide more examples where the product of two non-zero matrices is the zero matrix?
3. How do the eigenvalues of \( A \) and \( B \) relate to the product \( AB = 0 \)?
4. What are some practical applications of matrices in solving linear systems?
5. How does matrix multiplication differ from element-wise multiplication?

### Tip:
Always consider providing a counterexample when disproving mathematical statements, as it can clearly demonstrate the falsity of a general claim.

True or False? If A and B are n × n matrices with AB = 0, then A = 0 or B = 0.

Explore why the statement 'If A and B are n × n matrices with AB = 0, then A = 0 or B = 0' is false. This problem highlights core concepts in matrix algebra, such as matrix multiplication and counterexamples showing that neither matrix has to be zero.

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