https://file.aiquickdraw.com/mathbot/file/17299199717356vhl0rp9.jpg

Let's analyze each of the statements given in the image and determine whether they are true or false, along with justifications.

### Statement 11a
> If the equation \( Ax = 0 \) has only the trivial solution, then \( A \) is row equivalent to the \( n \times n \) identity matrix.

**True.**  
If \( Ax = 0 \) has only the trivial solution, this implies that \( A \) has full rank (i.e., all rows and columns are linearly independent). Therefore, \( A \) is invertible, and every invertible matrix is row equivalent to the identity matrix.

### Statement 11b
> If the columns of \( A \) span \( \mathbb{R}^n \), then the columns are linearly independent.

**True.**  
If the columns of \( A \) span \( \mathbb{R}^n \), \( A \) must have \( n \) linearly independent columns. This is because a spanning set in \( \mathbb{R}^n \) must be linearly independent to cover the entire \( n \)-dimensional space.

### Statement 11c
> If \( A \) is an \( n \times n \) matrix, then the equation \( Ax = b \) has at least one solution for each \( b \) in \( \mathbb{R}^n \).

**True.**  
If \( A \) is an \( n \times n \) matrix and has full rank (invertible), then \( Ax = b \) has a solution for any \( b \) in \( \mathbb{R}^n \) because \( A \) has an inverse. Therefore, there exists a unique solution for each \( b \).

### Statement 11d
> If the equation \( Ax = 0 \) has a nontrivial solution, then \( A \) has fewer than \( n \) pivot positions.

**True.**  
If \( Ax = 0 \) has a nontrivial solution, then \( A \) is not of full rank (i.e., it is not invertible and does not have a pivot in every row). Therefore, \( A \) has fewer than \( n \) pivot positions.

### Statement 11e
> If \( A^T \) is not invertible, then \( A \) is not invertible.

**True.**  
If \( A^T \) (the transpose of \( A \)) is not invertible, it implies that \( A \) is also not invertible, as the invertibility of \( A \) and \( A^T \) are equivalent. If one is non-invertible, the other must be as well.

### Summary of Answers
- **11a. True**
- **11b. True**
- **11c. True**
- **11d. True**
- **11e. True**

Would you like further details on any of these justifications or have additional questions? 

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### Additional Questions
1. What does it mean for a matrix to be row equivalent to the identity matrix?
2. Why does spanning \( \mathbb{R}^n \) imply linear independence for columns of an \( n \times n \) matrix?
3. What is the relationship between pivot positions and the rank of a matrix?
4. Can a non-square matrix have a unique solution for every \( b \) in \( \mathbb{R}^n \)?
5. How does the invertibility of \( A \) relate to the properties of \( A^T \)?

### Tip
Remember that the invertibility of a matrix is deeply connected to its rank: an \( n \times n \) matrix is invertible if and only if it has rank \( n \), meaning it must have linearly independent rows and columns.

Analyze each of the following statements to determine whether they are true or false:

11. a. If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix.
b. If the columns of A span R^n, then the columns are linearly independent.
c. If A is an n x n matrix, then the equation Ax = b has at least one solution for each b in R^n.
d. If the equation Ax = 0 has a nontrivial solution, then A has fewer than n pivot positions.
e. If A^T is not invertible, then A is not invertible.

Evaluate statements about matrix properties in linear algebra, including row equivalence to identity matrices, column independence, and the relationship between invertibility of A and A^T. This analysis covers important concepts like matrix rank, linear independence, and span in R^n. Suitable for undergraduate students in linear algebra.

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