Triangular and Trapezoidal Matrices

1. Any Matrix having Only Zero Values Either Below the Main Diagonal Or Above the Main Diagonal of the Matrix (Not Both) is called a Triangular Matrix (if it is a Square Matrix) or a Trapezoidal Matrix (if it is a Non-Square Matrix).
2. Any Matrix having Only Zero Values Below the Main Diagonal of the Matrix and Atleast One Non-Zero Value Above the Main Diagonal of the Matrix is called an Upper Triangular Matrix (if it is a Square Matrix) or a Upper Trapezoidal Matrix (if it is a Non-Square Matrix). Following are some examples of Upper Triangular/Trapezoidal Matrices
   
   \(\begin{bmatrix} 2 & 6 & 7\\ 0 & 3 & 1 \\ 0 & 0 & 13\end{bmatrix}\hspace{.5cm} \begin{bmatrix} 6 & 7\\ 0 & 2 \end{bmatrix} \hspace{.5cm} \begin{bmatrix} 1 & 3 & 7\\ 0 & 2 & -2\end{bmatrix} \hspace{.5cm} \begin{bmatrix} -2 & 9 \\ 0 & 3 \\ 0 & 0 \end{bmatrix}\)
   
   If All the elements of the Main Diagonal are also Zero then the Matrix is called Strictly Upper Triangular/Trapezoidal Matrix.
   
   If All the elements of the Main Diagonal are 1 then the Matrix is called Upper Uni-Triangular/Uni-Trapezoidal Matrix.
3. Any Matrix having Only Zero Values Above the Main Diagonal of the Matrix and Atleast One Non-Zero Value Below the Main Diagonal of the Matrix is called an Lower Triangular Matrix (if it is a Square Matrix) or a Lower Trapezoidal Matrix (if it is a Non-Square Matrix). Following are some examples of Lower Triangular/Trapezoidal Matrices
   
   \(\begin{bmatrix} 4 & 0 & 0\\ 3 & 7 & 0 \\ 9 & 11 & 5\end{bmatrix}\hspace{.5cm} \begin{bmatrix} 3 & 0\\ 1 & 2 \end{bmatrix} \hspace{.5cm} \begin{bmatrix} 2 & 0 & 0\\ 7 & -5 & 0\end{bmatrix} \hspace{.5cm} \begin{bmatrix} -5 & 0 \\ 7 & 1 \\ -4 & 8 \end{bmatrix}\)
   
   If All the elements of the Main Diagonal are also Zero then the Matrix is called Strictly Lower Triangular/Trapezoidal Matrix.
   
   If All the elements of the Main Diagonal are 1 then the Matrix is called Lower Uni-Triangular/Uni-Trapezoidal Matrix.
4. The Determinant of any Triangular Matrix is the Product of its Main Diagonal Elements.
5. The Transpose of any Lower Triangular/Trapezoidal Matrix is an Upper Triangular/Trapezoidal Matrix and vice versa.