given matrices A, B if for every i,j where b[i,j] is not 0 then the i-th column of A is 0. What can we know about AB

Given matrices \( A \) and \( B \), where for every \( i, j \) such that \( b[i, j] \) is not 0, the \( i \)-th column of \( A \) is 0, we can deduce that the matrix product \( AB \) will be the **zero matrix**.

### Explanation:
- Consider the matrix product \( AB \). The element at position \( (i, j) \) in the product \( AB \) is given by:
  \[
  (AB)[i, j] = \sum_{k} a[i, k] \cdot b[k, j]
  \]
  where \( a[i, k] \) and \( b[k, j] \) are the elements of matrices \( A \) and \( B \) respectively.

- Given that whenever \( b[k, j] \neq 0 \), the \( k \)-th column of \( A \) is entirely zero. This means that for any \( k \) where \( b[k, j] \) is non-zero, all elements in the \( k \)-th column of \( A \), including \( a[i, k] \), are zero.

- Thus, each term in the sum \( \sum_{k} a[i, k] \cdot b[k, j] \) is zero because either \( b[k, j] = 0 \), or if \( b[k, j] \neq 0 \), then \( a[i, k] = 0 \).

- Therefore, each entry \( (AB)[i, j] \) is zero, meaning the entire matrix \( AB \) is the zero matrix.

### Summary:
The product \( AB \) is the **zero matrix**.

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Would you like any additional details or have any questions? Here are some related questions to consider:

1. What are the conditions under which a matrix product results in the zero matrix?
2. How does the structure of matrix \( A \) affect the product \( AB \)?
3. What can be inferred if matrix \( A \) has no zero columns but \( AB \) is still the zero matrix?
4. Can matrix \( B \) be a zero matrix if \( AB \) is the zero matrix?
5. How would the conclusion change if \( b[i, j] = 0 \) for all \( i, j \)?
6. What if matrix \( B \) is an identity matrix, what does that imply about matrix \( A \)?
7. How does the rank of matrices \( A \) and \( B \) affect \( AB \)?
8. What happens if the columns of \( A \) are linearly dependent?

**Tip:** When dealing with matrix multiplication, analyzing the sparsity and structure of the matrices can provide insights into the resulting product without performing the entire multiplication.

Explore how matrices A and B interact in multiplication when the ith column of A is zero wherever b[i, j] is non-zero. Learn why the product AB results in the zero matrix under these conditions.

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