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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.

• Category:- Math
• Reference No.:- M9754

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