Project #8155 - Invertible matrices - logical equivalence

Write 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 Þ b, aÞ c, aÞ d, etc.However all statements must connect in some way.Example aÞ bÞ cÞ dÞ eÞ fÞ gÞ hÞ iÞ jÞ a.

Subject Mathematics
Due By (Pacific Time) 06/22/2013 09:00 pm
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