Question

Select four statements from the invertible matrix theorem and show that all four statements are true or false

This learning module introduced the invertible matrix theorem. This theorem is a collection of 12 equivalent statements. This means that when one statement in the theorem is true, all statements in the theorem are true. Similarly, when one statement in the theorem is false, all statements are false. For your initial post, propose a specific n x n matrix A, where n >= 3. Select four statements from the invertible matrix theorem and show that all four statements are true or false, depending on the matrix you selected. Make sure to clearly explain and justify your work. Also, make sure to label your statements using the notation used in the notes (e.g., part (a), part (f), etc.).
Theorem (The Invertible Matrix Theorem): Let A be an n x n matrix. Then the following statements are equivalent:
A is an invertible matrix.
A is row equivalent to the n x n identity matrix.
A has n pivot positions.
The equation Ax = 0 has only the trivial solution.
The columns of A form a linearly independent set.
The linear transformation x à Ax is one-to-one.
There is an n x n matrix C such that CA = I.
There is an n x n matrix D such that AD = I.AT is an invertible matrix.

Answer 1

A) A is an invertible matrix ( TRUE )

B) A is a row equivalent to the n x n identity matrix ( n = 3 ) ( TRUE )

C ) The equation Ax = 0 has only the trivial solution ( TRUE )

D ) The columns of A form a linearly independent set ( TRUE )

Explanation:

Assuming a matrix A

`\left[\begin{array}{ccc}1&2&1\\-1&0&3\\4&1&5\end{array}\right]`

det A = 1 [ 0 -3 ] + 2 [12 + 5 ] + 1[-1]

= -3 + 34 -1 = 30 ≠ 0

THEREFORE det A = 30 ≠ 0

Attached is the detailed solution of the given statements above