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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q23E (page 233)

If A and B are arbitrary $n 脳 n$ matrices, which of the matrices in Exercise 21 through 26 must be symmetric?
$A - A^{T}$ .

Short Answer

Expert verified

The matrix $A - A^{T}$ is not symmetric.

Step by step solution

01

Condition to be a symmetric.

A matrix is symmetric if and only if it is equal to its transpose.

All entries above the main diagonal of a symmetric matrix are reflected into equal entries below the diagonal. A matrix is skew-symmetric if and only if it is the opposite of its transpose.

For example,

$(\begin{matrix} 1 & 1 & - 1 \\ 1 & 2 & 0 \\ - 1 & 0 & 5 \end{matrix})$

02

Check AAΓ is symmetric or not.

Here it needs to find the matrix ${A_{A}}^{螕}$ .

To check that the matrix ${A_{A}}^{螕}$ is symmetric or not. Consider the terms below.

${(A - A^{T})}^{T} = A^{T} - {(A^{T})}^{T} = A^{T} - A = - (A - A^{T})$

So, the matrix need not to be symmetric unless $A - A^{T} = A^{T} - A$

For example, take

Let $A = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}]$

Now consider,

$A - A^{T} = [\begin{matrix} 0 & 1 \\ - 1 & 0 \end{matrix}] - [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] = [\begin{matrix} 0 & 2 \\ - 2 & 0 \end{matrix}]$

Which is not symmetric.

Thus, If A and B are arbitrary $n 脳 n$ matrices then the matrices $A - A^{T}$ is not symmetric.

Hence, the matrix is not symmetric.

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Most popular questions from this chapter

Consider an $m 脳 n$ matrix A with $K e r (A) = {\overset{鈫�}{0}}$ . Show that there exists an $n 脳 m$ matrix B such that $B A = l_{n}$ .

Consider an invertible n脳nmatrix A. Can you write A=RQ, where Ris an upper triangular matrix and Q is orthogonal?

In Exercises 40 through 46, consider vectors in ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}$ ; we are told thatrole="math" localid="1659495854834" ${\overset{鈫�}{v}}_{i}, {\overset{鈫�}{v}}_{j}$ is the entry $a_{i j}$ of matrix A.

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46. Find $p r o j_{v} ({\overset{鈫�}{v}}_{3})$ , where V =span role="math" localid="1659495997207" $({\overset{鈫�}{v}}_{1} . {\overset{鈫�}{v}}_{2})$ . Express your answer as a linear combination ofrole="math" localid="1659496026018" ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ .

Find scalar $a, b, c, d, e, f, g$ such that vectors $[\begin{matrix} a \\ d \\ f \end{matrix}], [\begin{matrix} b \\ 1 \\ g \end{matrix}], [\begin{matrix} c \\ \begin{matrix} e \\ \frac{1}{2} \end{matrix} \end{matrix}]$ are orthonormal.

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