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Found in: Page 191

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Question: Determine if the matrix pairs in Exercises 19-22 are controllable.20. $$A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)$$.

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

See the step by step solution

## Step 1: State the rank of a matrix

The rank of matrix $$A$$, denoted by rank$$A$$, is the dimension of the column space of $$A$$.

## Step 2: Determine the rank of the matrix

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.

## Step 3: Determine whether the matrix pairs are controllable

The pair $$\left( {A,B} \right)$$ is said to be controllable if 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.