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.

Ten of these equivalent statements are given below:

• A is an invertible matrix.

• A is row equivalent to the n × n identity matrix.

• A has n pivot positions.

• The equation Ax = 0 has only the trivial solution.

• The equation Ax = b has at least one solution for each b in R n .

• The columns of A span R n .

• The linear transformation x → Ax maps R n onto R n .

• There is an n × n matrix C such that CA = I.

• There is an n × n matrix D such that AD = I.

• The columns of A form a basis of R n .

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.

1. 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.
2. Provide an interpretation for the given statement: The n × n matrix A is invertible.

1. Explain what this statement means to you.

C. Write a brief essay ( suggested length of 1–2 pages ) in which you do the following:

1. Justify that the ten statements are logically equivalent to the statement "The n × n matrix A is invertible."
   Note: Your justification does not need to constitute a formal proof, and you may restrict your arguments to R 2 ( 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 A-1. (Do not use this justification, however, since it was provided as an example.)
2. Explain how each step in your justifications relates to your answers to parts A and B.

Price: $14.02

Solution: The downloadable solution consists of 4 pages, 1002 words.
Deliverable: Word Document