Converting matrices to upper triangle matrices without changing their determinant

Question

Is it possible to perform elementary row operations to every matrix in order to change its form into upper-triangle, without changing its determinant?

That is, by only adding multiples of other rows such that the determinant does not change. This would be of great help in for example proving that the determinant of a triangular block matrix is equal to the product of the determinants of its diagonal block matrices.

Answer 1

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.