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Chapter 4: Q5E (page 191)

If a \({\bf{3}} \times {\bf{8}}\) matrix A has a rank 3, find dim Nul A, dim Row A, and rank \({A^T}\).

Short Answer

Expert verified

5, 3, and 3

Step by step solution

01

Find dim Nul A

Using the rank theorem,you get:

\(\begin{aligned} {\rm{rank}}\,A + \dim \,{\rm{Nul}}A &= n\\3 + \dim \;{\rm{Nul}}\,A &= 8\\\dim \;{\rm{Nul}}\,A &= 8 - 3\\ &= 5\end{aligned}\)

02

Find dim row A

The dim row A is equal to the rank of A i.e., 3.

03

Find the rank of \({A^T}\)

\(\dim \,\;{\rm{Row}}\;A = \dim \,{\rm{Col}}\,{A^T} = 3\)

The rank of \({A^T}\) is equal to dim Col \({A^T}\); so the rank of \({A^T}\) is 3.

Thus, dim Nul A =5, dim row A=3, and the rank of \({A^T}\)=3.

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