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

19. \(A = \left( {\begin{array}{*{20}{c}}{.9}&1&0\\0&{ - .9}&0\\0&0&{.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right)\).

Short Answer

Expert verified

The matrix pairs \(\left( {A,B} \right)\) are controllable.

Step by step solution

01

Define the rank of a matrix

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

02

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:

\(\left( {\begin{array}{*{20}{c}}B&{AB}&{{A^2}B}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}0&1&0\\1&{ - .9}&{.81}\\1&{.5}&{.25}\end{array}} \right)\)

Perform an elementary row operation to produce the row-reduced echelon form of the matrix.

Interchange rows 1 and 2.

\( \sim \left( {\begin{array}{*{20}{c}}1&{ - .9}&{0.81}\\0&1&0\\1&{.5}&{.25}\end{array}} \right)\)

At row 3, subtract row 1 from row 3.

\( \sim \left( {\begin{array}{*{20}{c}}1&{ - .9}&{0.81}\\0&1&0\\0&{1.4}&{ - 0.56}\end{array}} \right)\)

At row 1, multiply row 2 by 0.9 and add it to row 1. At row 3, multiply row 2 by 1.4 and subtract it from row 3.

\( \sim \left( {\begin{array}{*{20}{c}}1&0&{0.81}\\0&1&0\\0&0&{ - 0.56}\end{array}} \right)\)

At row 3, multiply row 3 by \( - \frac{1}{{0.56}}\).

\( \sim \left( {\begin{array}{*{20}{c}}1&0&{0.81}\\0&1&0\\0&0&1\end{array}} \right)\)

At row 1, multiply row 3 by 0.81 and subtract it from row 1.

\( \sim \left( {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right)\)

The matrix has three pivot columns, so the rank of the matrix is 3.

03

Determine whether the matrix pairs are controllable

The pair \(\left( {A,B} \right)\) is said to becontrollable 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 3.

Thus, the matrix pairs \(\left( {A,B} \right)\) are controllable.

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Most popular questions from this chapter

Prove theorem 3 as follows: Given an \(m \times n\) matrix A, an element in \({\mathop{\rm Col}\nolimits} A\) has the form \(Ax\) for some x in \({\mathbb{R}^n}\). Let \(Ax\) and \(A{\mathop{\rm w}\nolimits} \) represent any two vectors in \({\mathop{\rm Col}\nolimits} A\).

  1. Explain why the zero vector is in \({\mathop{\rm Col}\nolimits} A\).
  2. Show that the vector \(A{\mathop{\rm x}\nolimits} + A{\mathop{\rm w}\nolimits} \) is in \({\mathop{\rm Col}\nolimits} A\).
  3. Given a scalar \(c\), show that \(c\left( {A{\mathop{\rm x}\nolimits} } \right)\) is in \({\mathop{\rm Col}\nolimits} A\).

Verify that rank \({{\mathop{\rm uv}\nolimits} ^T} \le 1\) if \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\5\end{array}} \right]\) and \({\mathop{\rm v}\nolimits} = \left[ {\begin{array}{*{20}{c}}a\\b\\c\end{array}} \right]\).

In Exercise 17, Ais an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if Ais not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

In Exercise 4, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

4. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{2}}\\{\bf{0}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\\{\bf{2}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{ - {\bf{7}}}\\{\bf{3}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{8}}\\{ - {\bf{7}}}\end{array}} \right)\)

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

  1. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} A\). (Hint: Explain why every vector in the column space of \(AB\) is in the column space of \(A\).
  2. Show that if \(B\) is \(n \times p\), then rank\(AB \le {\mathop{\rm rank}\nolimits} B\). (Hint: Use part (a) to study rank\({\left( {AB} \right)^T}\).)
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