Linear Algebra : Equivalence Statements Involving Elementary Matrices

Question

Prove : If an n*n matrix A can be expressed as a product of elementary matrices, Ax = b is consistent for every n*1 matrix b.

My thoughts on the question :

Since A can be expressed as a product of elementary matrices and elementary matrices are invertible, A can be expressed as a product of invertible matrices. Hence, A is invertible.

Now let us consider Ax = b

Since A is invertible and if A^-1 is it's inverse matrix,

(A^-1)Ax = (A^-1)b

x = (A^-1)b

Now how do we show that, Ax = b is consistent for every n*1 matrix b, in this case(when A is an n*n matrix which can be expressed as a product of elementary matrices)

Thank you!

Answer 1

You already proved it. Since A is the product of elementary matrices (they are invertible), then A is invertible. So for any nx1 vector b, the system Ax=b has the unique solution x=A^(-1) (b). As a result, the system Ax=b is consistent for each nx1 vector b.