10: With \({\mathop{\rm u}\nolimits} = \left( { - 2,3,1} \right)\) and A as in Exercise 8, determine if u is in Nul A.a

Matrix A in the form \(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{{{\mathop{\rm v}\nolimits} _3}}\end{array}} \right]\), as shown below:

\(A = \left[ {\begin{array}{*{20}{c}}{ - 3}&{ - 2}&0\\0&2&{ - 6}\\6&3&3\end{array}} \right]\)

The null spaceof matrix A is the set Nul Aof all solutions of the homogeneous equation\(Ax = 0\).

Calculate \(A{\mathop{\rm u}\nolimits} \), as shown below:

\(\begin{array}{c}A{\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}{ - 3}&{ - 2}&0\\0&2&{ - 6}\\6&3&3\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{ - 2}\\3\\1\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}{6 - 6 + 0}\\{0 + 6 - 6}\\{ - 12 + 9 + 3}\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}0\\0\\0\end{array}} \right]\end{array}\)

Since\[A{\mathop{\rm u}\nolimits} = 0\], \({\mathop{\rm u}\nolimits} \) is in Nul A.