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?

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.