https://file.aiquickdraw.com/mathbot/file/1727084826561q25zx21o.jpg

The matrix \( A \) given in the image is:

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

We are asked to check three conditions:
(i) Whether \( AA^T \) and \( A^T A \) are both symmetric.
(ii) Whether \( A + A^T \) is symmetric.
(iii) Whether \( A - A^T \) is skew-symmetric.

### Step-by-step analysis:

#### (i) \( AA^T \) and \( A^T A \) are both symmetric:
- A matrix \( M \) is symmetric if \( M = M^T \).
- First, we will calculate \( AA^T \) and check if it equals \( (AA^T)^T \).
- Then, we calculate \( A^T A \) and check if it equals \( (A^T A)^T \).

#### (ii) \( A + A^T \) is symmetric:
- A matrix is symmetric if \( A + A^T = (A + A^T)^T \).
- We'll compute \( A + A^T \) and verify whether it is symmetric.

#### (iii) \( A - A^T \) is skew-symmetric:
- A matrix is skew-symmetric if \( A - A^T = -(A - A^T)^T \).
- We'll compute \( A - A^T \) and check whether it satisfies this property.

Let me compute these matrices and verify the given conditions.### Results:

1. **\( AA^T \) and \( A^T A \) are both symmetric:**
   - \( AA^T = \begin{pmatrix} 11 & -26 & 16 \\ -26 & 98 & 29 \\ 16 & 29 & 146 \end{pmatrix} \) is symmetric.
   - \( A^T A = \begin{pmatrix} 11 & 26 & -16 \\ 26 & 98 & 29 \\ -16 & 29 & 146 \end{pmatrix} \) is symmetric.
   
   Therefore, **(i) is true**.

2. **\( A + A^T \) is symmetric:**
   - \( A + A^T = \begin{pmatrix} 2 & 0 & 0 \\ 0 & -10 & 16 \\ 0 & 16 & 18 \end{pmatrix} \), which is symmetric.
   
   Hence, **(ii) is true**.

3. **\( A - A^T \) is skew-symmetric:**
   - \( A - A^T = \begin{pmatrix} 0 & 6 & -2 \\ -6 & 0 & 0 \\ 2 & 0 & 0 \end{pmatrix} \), which satisfies the condition \( A - A^T = -(A - A^T)^T \).
   
   Therefore, **(iii) is true**.

All three conditions are satisfied.

---

Would you like further details or clarification on any step?

Here are 5 related questions:
1. What conditions must a matrix satisfy to be symmetric?
2. Can a non-square matrix have a symmetric transpose?
3. How are skew-symmetric matrices used in physics?
4. Can the product of two symmetric matrices be skew-symmetric?
5. What are the properties of orthogonal matrices in relation to symmetry?

**Tip:** Symmetric and skew-symmetric matrices have important applications in linear algebra, especially in decomposing matrices into simpler forms (like symmetric or skew-symmetric components).

If matrix A = [[1, 3, -1], [-3, -5, 8], [1, 8, 9]], then: (i) Are AA^T and A^T A both symmetric? (ii) Is A + A^T symmetric? (iii) Is A - A^T skew-symmetric?

Explore the symmetry properties of matrix A = [[1, 3, -1], [-3, -5, 8], [1, 8, 9]]. Learn how to verify if AA^T and A^T A are symmetric, if A + A^T is symmetric, and if A - A^T is skew-symmetric with detailed step-by-step solutions.

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