Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

20. \(A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)\).

Therank of matrix\(A\), denoted by rank\(A\), is the dimension of the column spaceof \(A\).

Calculate the rank of the matrix \(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}\end{array}} \right)\) to determine whether the matrix pair \(\left( {A,B} \right)\) is controllable.

Write the augmented matrix as shown below:

\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}1&{.5}&{.19}\\1&{.7}&{.45}\\0&0&0\end{array}} \right)\)

The matrix has 2 pivot columns, so the rank of the matrix must be less than 3.

The pair \(\left( {A,B} \right)\) is said to becontrollableif rank\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}& \cdots &{{A^{n - 1}}B}\end{array}} \right) = n\).

The rank of the matrix is less than 3.

Thus, the matrix pairs \(\left( {A,B} \right)\) are not controllable.