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Chapter 5: Q5.3-31E (page 267)

Question: Construct a nonzero \({\bf{2}} \times {\bf{2}}\) matrix that is invertible but not diagonalizable.

Short Answer

Expert verified

The matrix A is \(\left[ {\begin{array}{*{20}{c}}3&2\\0&3\end{array}} \right]\).

Step by step solution

01

Write the matrix which is invertible but not diagonalizable

Let the matrix of the order \(2 \times 2\) be:

\(A = \left[ {\begin{array}{*{20}{c}}5&2\\0&4\end{array}} \right]\)

The matrix A is triangular; therefore, the diagonal elements are the eigenvalues. So, the eigenvalues of A are 5 and 4

As the eigenvalues are distinct, therefore A is diagonalizable.

Consider a matrix with the same eigenvalues:

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

02

Apply characteristic equation for \(A\)

Apply the characteristic equation:

\(\begin{array}{c}\left( {A - 3I} \right){\bf{x}} = 0\\\left( {\left[ {\begin{array}{*{20}{c}}3&2\\0&3\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}3&0\\0&3\end{array}} \right]} \right)\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\end{array}} \right]\\\left[ {\begin{array}{*{20}{c}}0&2\\0&0\end{array}} \right]\left[ {\begin{array}{*{20}{c}}{{x_1}}\\{{x_2}}\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}0\\0\end{array}} \right]\end{array}\)

So, the eigenvector of the matrix is:

\(\begin{array}{c}{\bf{v}} = \left[ {\begin{array}{*{20}{c}}t\\0\end{array}} \right]\\ = \left[ {\begin{array}{*{20}{c}}1\\0\end{array}} \right]t\end{array}\)

Since the eigenvector is linearly dependent, therefore matrix A is invertible but not diagonalizable.

So, the matrix A is \(\left[ {\begin{array}{*{20}{c}}3&2\\0&3\end{array}} \right]\).

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Most popular questions from this chapter

Question: Construct a random integer-valued \(4 \times 4\) matrix \(A\).

  1. Reduce \(A\) to echelon form \(U\) with no row scaling, and use \(U\) in formula (1) (before Example 2) to compute \(\det A\). (If \(A\) happens to be singular, start over with a new random matrix.)
  2. Compute the eigenvalues of \(A\) and the product of these eigenvalues (as accurately as possible).
  3. List the matrix \(A\), and, to four decimal places, list the pivots in \(U\) and the eigenvalues of \(A\). Compute \(\det A\) with your matrix program, and compare it with the products you found in (a) and (b).

Assume the mapping\(T:{{\rm P}_2} \to {{\rm P}_{\bf{2}}}\)defined by \(T\left( {{a_0} + {a_1}t + {a_2}{t^2}} \right) = 3{a_0} + \left( {5{a_0} - 2{a_1}} \right)t + \left( {4{a_1} + {a_2}} \right){t^2}\) is linear. Find the matrix representation of\(T\) relative to the bases \(B = \left\{ {1,t,{t^2}} \right\}\).

Question: Diagonalize the matrices in Exercises \({\bf{7--20}}\), if possible. The eigenvalues for Exercises \({\bf{11--16}}\) are as follows:\(\left( {{\bf{11}}} \right)\lambda {\bf{ = 1,2,3}}\); \(\left( {{\bf{12}}} \right)\lambda {\bf{ = 2,8}}\); \(\left( {{\bf{13}}} \right)\lambda {\bf{ = 5,1}}\); \(\left( {{\bf{14}}} \right)\lambda {\bf{ = 5,4}}\); \(\left( {{\bf{15}}} \right)\lambda {\bf{ = 3,1}}\); \(\left( {{\bf{16}}} \right)\lambda {\bf{ = 2,1}}\). For exercise \({\bf{18}}\), one eigenvalue is \(\lambda {\bf{ = 5}}\) and one eigenvector is \(\left( {{\bf{ - 2,}}\;{\bf{1,}}\;{\bf{2}}} \right)\).

14. \(\left( {\begin{array}{*{20}{c}}4&0&{ - 2}\\2&5&4\\0&0&5\end{array}} \right)\)

For the Matrices A find real closed formulas for the trajectory x鈫�(t+1)=Ax鈫�(t)where x(0)鈫�=[01]

A=[43-34]

Question: In Exercises 21 and 22, \(A\) and \(B\) are \(n \times n\) matrices. Mark each statement True or False. Justify each answer.

  1. The determinant of \(A\) is the product of the diagonal entries in \(A\).
  2. An elementary row operation on \(A\) does not change the determinant.
  3. \(\left( {\det A} \right)\left( {\det B} \right) = \det AB\)
  4. If \(\lambda + 5\) is a factor of the characteristic polynomial of \(A\), then 5 is an eigenvalue of \(A\).
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