Chapter 2 - Matrices and Systems of Linear Equations - 2.8 The Invertible Matrix Theorem I - Problems - Page 189: 4

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Let $A$ and $B$ are invertible $n × n$ matrices By part (e) in the Invertible Matrix Theorem, we have: $$A=E_1E_2$$ $$B=E_3E_4$$ $E_k$ with $k=1,2,3,4...$ are elementary matrices So $$AB=E_1 E_2 E_3 E_4$$ is a product of elementary matrices too Since $E_k$ are invertible (by part a in the Invertible Matrix Theorem), $AB$ is vertible too (by Theorem 2.6.10)