prepare a brief essay (suggested length of 1–2 pages) in which you do the following:
Justify that the ten statements are logically equivalent to the statement “The n × n matrix A is invertible.”
(a) A is an invertible matrix.
(b) A is row equivalent to the n × n identity matrix.
(c) A has n pivot positions.
(d) The equation Ax = 0 has only the trivial solution.
(e) The equation Ax = b has at least one solution for each b in Rn
(f) The columns of A span Rn
(g) The linear transformation x → Ax maps Rn onto Rn.
(h) There is an n × n matrix C such that CA = I.
(i) There is an n × n matrix D such that AD = I.
(j) The columns of A form a basis of Rn.
This does not have to be a => b, a => c, a => d, etc. However all statements must connect in some way. ex a => b => c => d => e => f => g => h => i => j => a.