Determinant/Field/Upper triangular matrix/Fact

For an upper triangular matrix

${}M={\begin{pmatrix}b_{1}&\ast &\cdots &\cdots &\ast \\0&b_{2}&\ast &\cdots &\ast \\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&b_{n-1}&\ast \\0&\cdots &\cdots &0&b_{n}\end{pmatrix}}\,,$

we have

${}\det M=b_{1}b_{2}\cdots b_{n-1}b_{n}\,.$

In particular, for the identity matrix we get ${}\det E_{n}=1$.

Proof, Write another proof