Determinant of Triangular Matrix

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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:

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


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:

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

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):

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


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

Therefore:

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


By the defintion of the cofactor:

$T_{n + 1 \, n + 1} = (-1)^{n + 1 + n + 1} D_{n + 1 \, n + 1} = D_{n \, n}$

where $D_{n \, n}$ is the order $n$ determinant obtained from $D$ by deleting row $n + 1$ and column $n + 1$.


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

Therefore:

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

and the result follows by induction.


From:

Transpose of Upper Triangular Matrix is Lower Triangular

the result also holds by Determinant of Transpose for lower triangular matrices.

$\blacksquare$