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Triangular Matrices#

Definition: Upper Triangular Matrix (Right Triangular Matrix)

An upper or right triangular matrix is a Square Matrices which has non-zero entries only on its diagonal and above it.

\[ \begin{bmatrix}\ast & \ast & \cdots & \ast \\ 0 & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ast \\ 0 & \cdots & 0 & \ast\end{bmatrix} \]

Definition: Lower Triangular Matrix (Left Triangular Matrix)

A lower or left triangular matrix is a Square Matrices which has non-zero entries only on its diagonal and below it.

\[ \begin{bmatrix}\ast & 0 & \cdots & 0 \\ \ast & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ \ast & \cdots & \ast & \ast\end{bmatrix} \]

Theorem: Determinant

The determinant of a triangular matrix is the product of its diagonal entries.

Proof

TODO