Short Answer

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

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

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Step by Step Solution

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TABLE OF CONTENTS

Step 1: 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,

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

Thus, it gives

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

Therefore, if A and B are arbitrary $n \times n$ matrices then the matrices $A^{T} B A$ is not symmetric.

Hence, the matrix is not necessarily symmetric.

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Linear Algebra With Applications

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Short Answer

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

Step by Step Solution

Step 1: Condition to be a symmetric.

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If A and B are arbitrary n×n matrices, which of the matrices in Exercise 21 through 26 must be symmetric?AΓBA.

Step by Step Solution

Step 1: Condition to be a symmetric.

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