Math Problem Statement

Prove that an n × n matrix A is invertible if and only if it can be expressed as a product of elementary matrices.

Solution

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Matrix Inverses
Elementary Matrices

Formulas

A B = I_n
E_k E_{k-1} ... E_2 E_1 A = I_n
A = E_1^{-1} E_2^{-1} ... E_k^{-1}

Theorems

Matrix Invertibility Theorem
Elementary Matrices Theorem

Suitable Grade Level

Undergraduate (College Level)