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.