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Q15E

Expert-verifiedFound in: Page 191

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.

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

\({\rm{rank}}\,A \le 6\).

By **the rank theorem**, you get

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

Hence, the smallest possible dimension of Null *A* is 2.

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