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Found in: Page 93

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can you say about the columns of A?

The columns of A are linearly dependent.

See the step by step solution

## Step 1: The last column of AB is zero

Consider $${b_p}$$ to be the last column of B. The last column of $$AB$$ should be zero, according to the hypothesis.

## Step 2: Determine the columns of A

Here, $$A{b_p} = 0$$. On the other hand, B contains no column of zeros; therefore, $${b_p}$$ is not the zero vector. The equation $$A{b_p} = 0$$ represents the linear dependence relation among the columns of A.

Thus, the columns of A are linearly dependent.