Given: Within many proofs in mathematics, it is important to be able to demonstrate when two mathematical statements are logically equivalent to each other. There are a number of statements that are logically equivalent to the following: The n n matrix A is invertible. Nine of these equivalent statements are given below: 1. A is row equivalent to the n n identity matrix. 2. A has n pivot positions. 3. The equation Ax = 0 has only the trivial solution. 4. The equation Ax = b has at least one solution for each b in Rn. 5. The columns of A span Rn. 6. The linear transformation x Ax maps Rn onto Rn. 7. There is an n n matrix C such that CA = I. 8. There is an n n matrix D such that AD = I. 9. The columns of A form a basis of Rn. Task: Note: It is important that you think critically about your answer to parts A and B, since the remainder of this task depends on the quality of your response to these parts. A. Provide a definition for logical equivalence. Note: This definition should be structured so that it can be employed in the parts of the task that follow. B. Provide an interpretation for the given statement: The n n matrix A is invertible. 1. Explain what this statement means to you. C. Write an essay (suggested length of 23 pages) in which you do the following: 1. Prove that five of the other statements are logically equivalent to the statement The n n matrix A is invertible. Note: You may restrict your arguments to R2 (n = 2). For example, you can justify that matrix A has a non-zero determinant by noting that a zero determinant would introduce division by zero in the formula for finding the inverse of a 2 2 matrix. 2. Explain how each step in your justifications relates to your answers to parts A and B.

Given: Within many proofs in mathematics, it is important to be able to demonstrate when two mathematical statements are logically equivalent to each other. There are a number of statements that are logically equivalent to the following:

The n n matrix A is invertible.

Nine of these equivalent statements are given below:

1. A is row equivalent to the n n identity matrix.
2. A has n pivot positions.
3. The equation Ax = 0 has only the trivial solution.
4. The equation Ax = b has at least one solution for each b in Rn.
5. The columns of A span Rn.
6. The linear transformation x Ax maps Rn onto Rn.
7. There is an n n matrix C such that CA = I.
8. There is an n n matrix D such that AD = I.
9. The columns of A form a basis of Rn.

Task:

Note: It is important that you think critically about your answer to parts A and B, since the remainder of this task depends on the quality of your response to these parts.

A. Provide a definition for logical equivalence.

Note: This definition should be structured so that it can be employed in the parts of the task that follow.

B. Provide an interpretation for the given statement: The n n matrix A is invertible.
1. Explain what this statement means to you.

C. Write an essay (suggested length of 2-3 pages) in which you do the following:
1. Prove that five of the other statements are logically equivalent to the statement The n n matrix A is invertible.

Note: You may restrict your arguments to R2 (n = 2). For example, you can justify that matrix A has a non-zero determinant by noting that a zero determinant would introduce division by zero in the formula for finding the inverse of a 2 2 matrix.

2. Explain how each step in your justifications relates to your answers to parts A and B.