Math Problem Statement

think really hard of an example of the conditions I gave you where AB is not the zero matri

Solution

Let's think through this carefully. The conditions state:

  • For every i,ji, j where b[i,j]b[i, j] is not 0, the ii-th column of AA is 0.

Given this, I initially concluded that ABAB would always be the zero matrix. However, let's explore if there's any scenario where ABAB might not be the zero matrix.

Analysis:

  1. Condition Restatement:

    • If b[i,j]b[i, j] is non-zero, the ii-th column of AA is zero.
    • This implies that every non-zero element in matrix BB corresponds to a column in AA that is entirely zero.
  2. Consequences of the Condition:

    • If BB has a non-zero element b[i,j]b[i, j], the entire ii-th column of AA is zero.
    • The contribution to the jj-th column of the product ABAB from the ii-th column of AA will be zero because the corresponding elements in AA are all zero.

Revisiting the Possibility:

Consider a special case where matrix BB is entirely zero.

  • In this case, there are no non-zero elements in BB, so the condition on the columns of AA doesn't force any column of AA to be zero.
  • Here, ABAB will be the zero matrix simply because BB is zero, not due to the interaction of non-zero elements and zero columns in AA.

Possible Non-Zero ABAB?

To achieve ABAB being non-zero, we'd have to break the initial condition—there seems to be no way to construct AA and BB satisfying the condition where ABAB is non-zero.

Thus, under the given condition, ABAB will always be the zero matrix.

Would you like to explore this further or discuss related concepts? Here are some related questions to consider:

  1. Can you construct a different scenario where ABAB might be non-zero without violating any conditions?
  2. How does the sparsity pattern in matrix BB influence the structure of AA under these conditions?
  3. What happens if matrix AA has a specific pattern, like being diagonal or upper triangular?
  4. Is there a general theorem that covers conditions like this in linear algebra?
  5. How do these conditions relate to the concept of matrix rank?
  6. What role do matrix norms play in analyzing the product ABAB?
  7. How might these conditions be visualized using block matrices?
  8. What implications does this have for solving linear systems involving matrices AA and BB?

Tip: In linear algebra, carefully considering each condition and its implications can help identify potential edge cases or confirm that a certain result holds universally.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Multiplication
Zero Matrix

Formulas

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Theorems

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Suitable Grade Level

Advanced Undergraduate