Converting matrices to upper triangle matrices without changing their determinant

Short answer, yes.

According to the definition of matrix determinant : wiki $$ \det[a_1, ... , b\cdot a_j + c\cdot v, ... , a_n] = b\det[a_1, ... , a_j, ... , a_n] + c\det[a_1, ... , v, ... , a_n] $$

if $b=1$ and $v=a_k\{\text{with } j\neq k\}$ then,

• obviously that is a basic column operation.
• it does not change the determinant, since $\det[a_1, ...., a_k, ..., a_k,..., a_n] =0 $

The idea can easily also be applied to row operations. So a "valid" row operation that does not change the determinant of the matrix is $$ a_j' = a_j + \alpha \cdot a_k \quad (k\neq j) $$

The second part of the problem is how to use such row operations to get to a upper triangular matrix. It seems obvious to me, but I don't have a succinct way to write it down. Hope you can figure that out yourself.