Question 1: Identifying Symmetric Matrices

A matrix $A$ is symmetric if $A = A^T$, meaning that the matrix is equal to its transpose. The transpose of a matrix is obtained by swapping its rows and columns.

Let’s analyze each matrix:

1. Matrix 1: $9ij$

   • This matrix is a function of $i$ and $j$. However, without more specific values or context for $i$ and $j$, we cannot confirm whether it is symmetric.

2. Matrix 2: $9i8 + 5j7$

   • This matrix appears to involve terms dependent on both $i$ and $j$. It seems unlikely that this would satisfy $A = A^T$ unless specific conditions are imposed on $i$ and $j$, so it is probably not symmetric.

3. Matrix 3: $i8 + j8$

   • This matrix is a sum of terms involving $i$ and $j$. However, unless the specific functional form of $i$ and $j$ guarantees that $A = A^T$, we cannot automatically assume it is symmetric.

4. Matrix 4: $9i8j7$

   • Again, the values depend on both $i$ and $j$, and without additional information, it seems unlikely that this matrix would be symmetric.

Thus, none of the matrices listed appear to be symmetric, based on the given information.

Question 2: Solutions to $A \mathbf{x} = 0$ and $A \mathbf{x} = \mathbf{b}$

• If $A \mathbf{x} = 0$ has only the trivial solution, it implies that matrix $A$ is invertible (i.e., non-singular).
• When $A$ is invertible, the equation $A \mathbf{x} = \mathbf{b}$ has exactly one solution for each $\mathbf{b}$, as the inverse matrix $A^{-1}$ can be used to solve the system by $\mathbf{x} = A^{-1} \mathbf{b}$.

Thus, the statement is True.

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Would you like to review any details or have further questions on the solutions?

Here are some related questions to consider:

1. What is the definition of a symmetric matrix?
2. How do you compute the transpose of a matrix?
3. What conditions make a matrix invertible?
4. How can you prove a matrix has only the trivial solution for $A \mathbf{x} = 0$?
5. How does the determinant of a matrix relate to its invertibility?

Tip: For a matrix to be symmetric, it must have identical elements across the diagonal, meaning $A_{ij} = A_{ji}$.