# Matrix with unit determinant as a product of elementary matrices.

There are three types of elementary matrices:

• Type 1: matrices obtained by interchanging the ith row of $$I$$ and jth row of $$I$$;

• Type 2: matrices obtained by multiplying the ith row of $$I$$ by $$\lambda\neq 0$$;

• Type 3: matrices obtained by adding $$\lambda$$ times the jth row of $$I$$ to the ith row of $$I$$.

I know that every invertible matrix $$A$$ can be written as a product of elementary matrices (of types 1, 2 and 3). According to Example 12.2.5 here, it is clear that

(a) if $$\det A=1$$ then $$A$$ can be written as a product of elementary matrices of type 2 and type 3.

(b) if $$\det A=1$$ and the factorization of $$A$$ (in terms of elementary matrices) has a matrix of type 2 with some $$\lambda$$, than this factorization also has a matrix of type 2 with $$\lambda^{-1}$$.

The item (b) is true because otherwise we would have $$\det A\neq 1$$. However, I can't understand why (a) is true. So, I'd like help to prove it.

Thanks.

• For me, the first implication of your hint is: the number $k$ of matrices of type 1 in the factorization of $A$ is even. Why $k>0$ is not possible? – Pedro May 17 '15 at 23:58