Consider a matrix A with entries defined as indicated below.  Which of these are symmetric?  Select ALL that apply.





9 i j9 i j

9 i8+5 j79 i8+5 j7

i8+j8i8+j8

9 i8 j79 i8 j7

Question 2

1 point

If *A***x**=**0** has only the trivial solution, then *A***x**=**b** has exactly one solution for each **b**.





True

False

### 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 \( A_{ij} = A_{ji} \).

Consider a matrix A with entries defined as indicated below. Which of these are symmetric? Select ALL that apply. 9i j, 9i 8+5j7, i8+j8, 9i8 j7. If A * x = 0 has only the trivial solution, then A * x = b has exactly one solution for each b. True or False?

Learn how to identify symmetric matrices by examining the transpose of a matrix and determine the conditions under which a matrix equation A * x = 0 has only the trivial solution. This problem covers key concepts from linear algebra, including matrix symmetry and invertibility.

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