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Let's solve part (ii) of the problem step by step:

We are given matrix \( A \) and asked to show that \( A + A^T \) is a symmetric matrix.

### Step 1: Recall the Definition of Symmetric Matrix
A matrix \( M \) is symmetric if \( M = M^T \). That is, a matrix is symmetric if it is equal to its own transpose.

### Step 2: Write Down the Matrix \( A \)

The given matrix \( A \) is:
\[
A = \begin{pmatrix}
1 & 3 & -1 \\
-3 & -5 & 8 \\
1 & 8 & 9
\end{pmatrix}
\]

### Step 3: Compute the Transpose of \( A \), Denoted by \( A^T \)

To find the transpose \( A^T \), we swap the rows and columns of \( A \). The result is:

\[
A^T = \begin{pmatrix}
1 & -3 & 1 \\
3 & -5 & 8 \\
-1 & 8 & 9
\end{pmatrix}
\]

### Step 4: Add \( A \) and \( A^T \)

Now, we compute \( A + A^T \). This is done by adding the corresponding elements of \( A \) and \( A^T \).

\[
A + A^T = \begin{pmatrix}
1 & 3 & -1 \\
-3 & -5 & 8 \\
1 & 8 & 9
\end{pmatrix} + \begin{pmatrix}
1 & -3 & 1 \\
3 & -5 & 8 \\
-1 & 8 & 9
\end{pmatrix}
\]

Perform the element-wise addition:

\[
A + A^T = \begin{pmatrix}
1 + 1 & 3 + (-3) & -1 + 1 \\
-3 + 3 & -5 + (-5) & 8 + 8 \\
1 + (-1) & 8 + 8 & 9 + 9
\end{pmatrix}
\]

This simplifies to:

\[
A + A^T = \begin{pmatrix}
2 & 0 & 0 \\
0 & -10 & 16 \\
0 & 16 & 18
\end{pmatrix}
\]

### Step 5: Check if \( A + A^T \) is Symmetric

For \( A + A^T \) to be symmetric, it must satisfy the condition \( (A + A^T) = (A + A^T)^T \). In other words, the matrix must be equal to its transpose.

Let's take the transpose of \( A + A^T \):

\[
(A + A^T)^T = \begin{pmatrix}
2 & 0 & 0 \\
0 & -10 & 16 \\
0 & 16 & 18
\end{pmatrix}^T
\]

Since the transpose of a symmetric matrix is itself, we find:

\[
(A + A^T)^T = \begin{pmatrix}
2 & 0 & 0 \\
0 & -10 & 16 \\
0 & 16 & 18
\end{pmatrix}
\]

Clearly, \( A + A^T = (A + A^T)^T \), confirming that \( A + A^T \) is indeed symmetric.

### Final Answer:

Yes, \( A + A^T \) is a symmetric matrix.

---

Do you want details or have any questions? Here are 5 relative questions for more practice:

1. What is the condition for a matrix to be skew-symmetric?
2. How do you compute the determinant of a 3x3 matrix?
3. If \( A \) is a symmetric matrix, what can be said about its eigenvalues?
4. Can a non-square matrix be symmetric or skew-symmetric?
5. How is matrix multiplication affected when using symmetric matrices?

**Tip**: To check if a matrix is symmetric or skew-symmetric, always compute its transpose and compare it to the original matrix!

If A = [[1, 3, -1], [-3, -5, 8], [1, 8, 9]], prove that A + A^T is a symmetric matrix.

Solve a matrix algebra problem where matrix A + A^T is proven to be symmetric by using the transpose of matrix A. The solution includes a detailed step-by-step calculation and explanation of symmetry in matrices.

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