Determinant of Triangular Matrix

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

Let $\det \left({\mathbf T_n}\right)$ be the determinant of $\mathbf T_n$.

Then $\det \left({\mathbf T_n}\right)$ is equal to the product of all the diagonal elements of $\mathbf T_n$.

That is:
 * $\displaystyle \det \left({\mathbf T_n}\right) = \prod_{k \mathop = 1}^n a_{kk}$

Proof
Let $\mathbf T_n$ be an upper triangular matrix of order $n$. We proceed by induction on $n$, the number of rows of $\mathbf T_n$.

Basis for the Induction
For $n = 1$, the determinant is $a_{11}$, which is clearly also the diagonal element.

This forms the basis for the induction.

Induction Hypothesis
Fix $n \in \N$. Then, let $\mathbf T_n = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ 0 & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \\ \end{bmatrix}$ be an upper triangular matrix.

Assume that


 * $\displaystyle \det \left({\mathbf T_n}\right) = \prod_{k \mathop = 1}^n a_{kk}$

This forms our induction hypothesis.

Induction Step
Let $\mathbf T_{n+1}$ be an upper triangular matrix of order $n+1$. Then, by the Expansion Theorem for Determinants (expanding across the $n+1$th row):


 * $\displaystyle D = \det \left({\mathbf T_{n+1}}\right) = \sum_{k \mathop = 1}^{n+1} a_{n+1\,k} A_{n+1\,k}$

Because $\mathbf T_{n+1}$ is upper triangular, $a_{n+1\,k} = 0$ when $k < n + 1$. Therefore,


 * $\displaystyle \det \left({\mathbf T_{n+1}}\right) = a_{n+1\,n+1} A_{n+1\,n+1}$

By the defintion of the cofactor, $A_{n+1\,n+1} = (-1)^{n+1 + n+1} D_{n+1\,n+1} = D_{n+1\,n+1}$, where $D_{n+1\,n+1}$ is the order $n$ determinant obtained from $D$ by deleting row $n+1$ and column $n+1$.

But $D_{n+1\,n+1}$ is just the determinant of an upper triangular matrix $\mathbf T_n$. Therefore,


 * $\displaystyle \det \left({\mathbf T_{n+1}}\right) = a_{n+1\,n+1} \det \left({\mathbf T_n}\right)$

and the result follows by induction.

Because the transpose of a lower triangular matrix is upper triangular, the result follows by Determinant of Transpose for lower triangular matrices as well.