An n ×n matrix 𝐀 is called symmetric if 𝐀^T=𝐀 ; that is, if ai j=aj i, for all i, j=1, …, n . Sh

step 1 In this video, we're going to go through the answer to question number 41 from chapter 9 .3. So

An n ×n matrix 𝐀 is called symmetric if 𝐀^T=𝐀 ; that is, if...

An $n \times n$ matrix $\mathbf{A}$ is called symmetric if $\mathbf{A}^{T}=\mathbf{A} ;$ that
is, if $a_{i j}=a_{j i},$ for all $i, j=1, \ldots, n .$ Show that if $\mathbf{A}$ is an $n \times n$ matrix, then $\mathbf{A}+\mathbf{A}^{T}$ is a symmetric matrix.

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Key Concepts

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its own transpose. This means that the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column for all indices. This property is crucial when considering operations that depend on the alignment of indices, such as matrix addition and eigenvalue computations.

Matrix Transpose

The transpose of a matrix is an operation that flips the matrix over its diagonal, converting the rows of the matrix into columns and vice versa. It is a fundamental concept in linear algebra that is used to express symmetry and plays a pivotal role in the analysis of matrix properties and operations.

Matrix Addition

Matrix addition is the operation of adding two matrices by adding corresponding elements from each matrix. It is key to many proofs and computations in linear algebra, particularly when demonstrating properties that are preserved under addition, such as the symmetry in the sum of a matrix and its transpose.

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