This is the exact question/instructions I am to follow. I answered part (a) and found the matrix I used is not invertible. Now I am stuck.
propose a specific $n \times n$ matrix $A$, where $n \ge 3$. 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.). In responding to your classmates’ posts, verify that their work is correct and select three additional statements they did not originally address and show that these statements are true (or false). Again, make sure to show your work.
My matrix: $\pmatrix{1& 1& 1\\ 1& 1& 0\\ 0& 0& 1}$
I'm trying to answer parts a,b,c, and k of:
Let A be a square n x n matrix. Then, the following statements are equivalent. That is, for a given A, the statements are either all true or all false.
- a. A is an invertible matrix.
- b. A is row equivalent to the n x n identity matrix.
- c. A has n pivot positions
- d. The equation Ax = 0 has only the trivial solution.
- e. The columns of A for a linearly independent set.
- f. The linear transformation x ↦ Ax is one-to-one.
- g. The equation Ax = b has at least one solution for each b in ℝn.
- h. The columns of A span ℝn.
- i. The linear transformation x ↦ Ax maps ℝn onto ℝn.
- j. There is an n x n matrix C such that CA = I.
- k. There is an n x n matrix D such that AD = I.
- l. AT is an invertible matrix.