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Chapter 7: Q3E (page 395)

Determine which of the matrices in Exercises 1鈥�6 are symmetric.

3. \(\left( {\begin{aligned}{{}}2&{\,\,3}\\{\bf{2}}&4\end{aligned}} \right)\)

Short Answer

Expert verified

The given matrix is not symmetric.

Step by step solution

01

Find the transpose

A matrix\(A\) with, \(n \times n\) dimension, is symmetric if it satisfies the equation\({A^T} = A\).

It is given that\(A = \left( {\begin{aligned}{{}}2&{\,\,3}\\2&4\end{aligned}} \right)\). Find the transpose of\(A\), as shown below:

\(\begin{aligned}{}{A^T} = \left( {\begin{aligned}{{}}2&2\\3&4\end{aligned}} \right)\\ \ne A\end{aligned}\)

02

Draw the conclusion

As \({A^T} \ne A\), so it can be concluded that \(A\) is not asymmetric matrix.

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Most popular questions from this chapter

Question: 14. Exercises 12鈥�14 concern an \(m \times n\) matrix \(A\) with a reduced singular value decomposition, \(A = {U_r}D{V_r}^T\), and the pseudoinverse \({A^ + } = {U_r}{D^{ - 1}}{V_r}^T\).

Given any \({\rm{b}}\) in \({\mathbb{R}^m}\), adapt Exercise 13 to show that \({A^ + }{\rm{b}}\) is the least-squares solution of minimum length. [Hint: Consider the equation \(A{\rm{x}} = {\rm{b}}\), where \(\mathop {\rm{b}}\limits^\^ \) is the orthogonal projection of \({\rm{b}}\) onto Col \(A\).

Classify the quadratic forms in Exercises 9鈥�18. Then make a change of variable, \({\bf{x}} = P{\bf{y}}\), that transforms the quadratic form into one with no cross-product term. Write the new quadratic form. Construct \(P\) using the methods of Section 7.1.

11. \({\bf{2}}x_{\bf{1}}^{\bf{2}} - {\bf{4}}{x_{\bf{1}}}{x_{\bf{2}}} - x_{\bf{2}}^{\bf{2}}\)

10.Determine which of the matrices in Exercises 7鈥�12 are orthogonal. If orthogonal, find the inverse.

10. \(\left( {\begin{aligned}{{}}{1/3}&{\,\,2/3}&{\,\,2/3}\\{2/3}&{\,\,1/3}&{ - 2/3}\\{2/3}&{ - 2/3}&{\,\,1/3}\end{aligned}} \right)\)

All symmetric matrices are diagonalizable.

In Exercises 25 and 26, mark each statement True or False. Justify each answer.

a. An\(n \times n\)matrix that is orthogonally diagonalizable must be symmetric.

b. If\({A^T} = A\)and if vectors\({\rm{u}}\)and\({\rm{v}}\)satisfy\(A{\rm{u}} = {\rm{3u}}\)and\(A{\rm{v}} = {\rm{3v}}\), then\({\rm{u}} \cdot {\rm{v}} = {\rm{0}}\).

c. An\(n \times n\)symmetric matrix has n distinct real eigenvalues.

d. For a nonzero \({\rm{v}}\) in \({\mathbb{R}^n}\) , the matrix \({\rm{v}}{{\rm{v}}^T}\) is called a projection matrix.
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