To identify a symmetric matrix, you need to check whether the matrix is equal to its transpose. A symmetric matrix is a square matrix that satisfies the following condition:

$A = A^T$

where $A^T$ is the transpose of matrix $A$.

Definition of Transpose

The transpose of a matrix $A$ is obtained by flipping the matrix over its diagonal. This means that the element in the $i$-th row and $j$-th column of the matrix becomes the element in the $j$-th row and $i$-th column in the transpose.

Symmetric Matrix Condition

A matrix $A = [a_{ij}]$ is symmetric if $a_{ij} = a_{ji}$ for all $i$ and $j$. In other words, each element in the matrix should be equal to the corresponding element about the main diagonal.

Example

Let's consider a 3x3 matrix $A$:

1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$ To determine if this matrix is symmetric, let's find its transpose $$A^T$$: $$A^T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$ Notice that the transpose of $$A$$ is identical to the original matrix $$A$$. This means that: $$A = A^T$$ Thus, the matrix $$A$$ is symmetric. ### Non-Example Consider another matrix $$B$$: $$B = \begin{bmatrix} 1 & 7 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$ The transpose of $$B$$, $$B^T$$, is: $$B^T = \begin{bmatrix} 1 & 2 & 3 \\ 7 & 4 & 5 \\ 3 & 5 & 6 \end{bmatrix}$$ Here, $$B \neq B^T$$ because the element $$b_{12} = 7$$ does not equal $$b_{21} = 2$$. Therefore, matrix $$B$$ is **not** symmetric. ### Summary To identify a symmetric matrix: 1. The matrix must be square (same number of rows and columns). 2. The matrix must be equal to its transpose, $$A = A^T$$. Would you like more details or have any questions? ### Related Questions: 1. What is the transpose of a matrix, and how do you compute it? 2. Can a non-square matrix be symmetric? Why or why not? 3. What are some properties of symmetric matrices in linear algebra? 4. How does the concept of a symmetric matrix relate to other types of matrices, like skew-symmetric matrices? 5. What are some applications of symmetric matrices in mathematics or physics? **Tip:** When checking for symmetry, always ensure the matrix is square first. Symmetry is only defined for square matrices.