Let E1 Ek be n times n elementary matrices and let A E1Ek

Let E_1, ...., E_k be n times n elementary matrices and let A = E_1...E_k. Explain why Ax rightarrow = b rightarrow has a unique solution for all n times 1 matrix b rightarrow.

Solution

All elemnatry matrices are invertible and product of invertible matrices is also invertible

HEnce, A is invertible

For any b we can write

x=A^{-1}b

as solution to

Ax=b