c).      i).  How do you identify a symmetric matrix? Explain with an example.

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 \(A\):

\[
A = \begin{bmatrix}
1 & 2 & 3 \\
2 & 4 & 5 \\
3 & 5 & 6
\end{bmatrix}
\]

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.

Learn how to identify symmetric matrices in linear algebra. Understand the condition where a matrix equals its transpose and explore examples and non-examples. Suitable for advanced high school students and anyone interested in matrix theory.

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