Question

If a\({\bf{6}} \times {\bf{3}}\)matrix A has a rank 3, find dim Nul A, dim Row A, and rank\({A^T}\).

Short answer

0, 3, and 3

Step-by-step solution

Find dim Nul A

Using the rank theorem,you get:

\(\begin{aligned} {\rm{rank}}\,A + \dim \,{\rm{Nul}}A &= n\\3 + \dim \;{\rm{Nul}}\,A &= 3\\\dim \;{\rm{Nul}}\,A &= 3 - 3\\ &= 0\end{aligned}\)

Find dim row A

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

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 =0, dim row A=3, and the rank of \({A^T}\)=3.