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What can be concluded about the columns of matrix A if the last column of the product AB is entirely zeros, but B itself has no column of zeros?

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Final answer:

When the last column of the product AB is all zeros and B has no column of zeros, it can be concluded that the columns of matrix A are linearly dependent.

Step-by-step explanation:

When the last column of the product AB is entirely zeros but B itself has no column of zeros, we can conclude that the columns of matrix A are linearly dependent. This means that at least one column of A can be expressed as a linear combination of the other columns.

This can be understood by considering the product AB. The columns of AB are formed by multiplying each column of A with each corresponding column of B and then summing them. If the last column of AB is all zeros, it means that the linear combination of the columns of A that corresponds to the last column of AB is equal to zero. This implies that the columns of A must be linearly dependent.

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