Determinant of Triangular Matrix

Theorem
Let $\mathbf T_n$ be a triangular matrix (either upper or lower) of order $n$.

Let $\map \det {\mathbf T_n}$ be the determinant of $\mathbf T_n$.

Then $\map \det {\mathbf T_n}$ is equal to the product of all the diagonal elements of $\mathbf T_n$.

That is:
 * $\displaystyle \map \det {\mathbf T_n} = \prod_{k \mathop = 1}^n a_{k k}$

Proof
Let $\mathbf T_n$ be an upper triangular matrix of order $n$.

From Determinant of Upper Triangular Matrix, the determinant of $\mathbf T_n$ is equal to the product of all the diagonal elements of $\mathbf T_n$.

From Transpose of Upper Triangular Matrix is Lower Triangular, the transpose $\mathbf T_n^\intercal$ of $\mathbf T_n$ is a lower triangular matrix.

From Determinant of Transpose, the determinant of $\mathbf T_n^\intercal$ equals the determinant of $\mathbf T_n$.