select four statements from the invertible matrix theorem and show that all four statements are true or false
this learning module introduced the invertible matrix theorem. this theorem is a collection of 12 equivalent statements. this means that when one statement in the theorem is true, all statements in the theorem are true. similarly, when one statement in the theorem is false, all statements are false. for your initial post, propose a specific n x n matrix a, where n > = 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),
theorem (the invertible matrix theorem): let a be an n x n matrix. then the following statements are equivalent:
a is an invertible matrix.
a is row equivalent to the n x n identity matrix.
a has n pivot positions.
the equation ax = 0 has only the trivial solution.
the columns of a form a linearly independent set.
the linear transformation x à ax is one-to-one.
the equation ax = b has at least one solution for each b in pastedimage_ohc3rrwkynqmcj5zzf3co6f lld6y.
the columns of a span pastedimage_pzfbc6caq6376idt98x6kbg bja3r.
the linear transformation x à ax maps pastedimage_q19dyrqa66w6pzkvd6qntye ubll3 onto pastedimage_dmc0illlvzpimrjuv5soz2g 0p5cb.
there is an n x n matrix c such that ca = i.
there is an n x n matrix d such that ad = i.
at is an invertible matrix.