# Determinant of Triangular Matrix

## Theorem

### Determinant of Upper Triangular Matrix

Let $\mathbf T_n$ be an upper triangular matrix 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:

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

### Determinant of Lower Triangular Matrix

Let $\mathbf T_n$ be a lower triangular matrix 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:

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