Confusing about Matrix A is invertibe?

Question

Let A be an n×n matrix of real or complex numbers. Show that matrix A is invertible if which of the following satement are correct

a) The columns of A are linearly independent.

b) The columns of A span R^n . c) The rows of A are linearly independent.

d) The kernel of A is 0.

e) The only solution of the homogeneous equations Ax = 0 is x = 0.

f) The linear transformation TA : R^n → R n defined by A is 1-1. g) The linear transformation TA : R^n → R^n defined by A is onto. h) The rank of A is n.

THIS is the orginal question

THIS is the orginal question

I think a b c d e f g h i j all are correct...

Answer 1

Each of those conditions is equivalent to the invertibility of $A $. Any linear algebra text should include a proof of (most of) those, or have them as exercises (as they are all fairly straightforward).