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Expert-verified Found in: Page 191 ### 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 # If A is a $${\bf{6}} \times {\bf{8}}$$ matrix, what is the smallest possible dimension of Null A?

The smallest possible dimension of Null A is 2.

See the step by step solution

## Step 1: Describe the given data

From the given $$6 \times 8$$ matrix, the number of pivots cannot exceed 6. That is

$${\rm{rank}}\,A \le 6$$.

## Step 2: Use the rank theorem

By the rank theorem, you get

\begin{aligned} n &= {\rm{rank}}\,A + \dim \,{\rm{Null}}\,A\\8 &\le 6 + \dim \,{\rm{Null}}\,A\\8 - 6 &\le \dim \,{\rm{Null}}\,A\\2 &\le \dim \,{\rm{Null}}\,A.\end{aligned}

## Step 3: Draw a conclusion

Hence, the smallest possible dimension of Null A is 2. ### Want to see more solutions like these? 