Question

What is a strictly triangular matrix?

Answer 1

Hint: The sum of two upper triangular matrices is upper triangular. The product of two upper triangular matrices is upper triangular. The inverse of an upper triangular matrix, where extant, is upper triangular. The product of an upper triangular matrix and a scalar is upper triangular.

Complete step by step solution:
In linear algebra, a triangular matrix is a special type of square matrix. If all the elements above the diagonal of a square matrix are zero, then it is called an upper triangular matrix. If all the elements below the diagonal of a square matrix are zero, then it is called a lower triangular matrix. Similarly, when all the elements on the diagonal of a square triangular matrix (may be upper or lower triangular) are 0, then it is called a strictly triangular (strictly upper or lower) matrix.
As mentioned, a square matrix is called upper triangular if all the entries below the main diagonal are zero and an upper or right triangular matrix is commonly denoted with the variable U matrix that is both upper and lower triangular is diagonal as shown:
\[U = \left( {\begin{array}{*{20}{c}}
  {{u_{1,1}}}& \ldots &{{u_{1,n}}} \\
   \vdots & \ddots & \vdots \\
  0& \cdots &{{a_{n,n}}}
\end{array}} \right)\]
A strictly upper triangular matrix is an upper triangular matrix having 0s along the diagonal as well as the lower portion i.e., a matrix\[A = \left[ {{a_{i,j}}} \right]\]for \[{a_{i,j}} = 0\] and \[i \geqslant j\].
\[U = \left( {\begin{array}{*{20}{c}}
  0&{{a_{1,2}} \ldots }&{{u_{1,n}}} \\
   \vdots & \ddots & \vdots \\
  0&{0 \cdots }&0
\end{array}} \right)\]
A square matrix is called a lower triangular if all the entries above the main diagonal are zero. A lower or left triangular matrix is commonly denoted with the variable L, and is of the from:
\[L = \left( {\begin{array}{*{20}{c}}
  {{l_{1,1}}}& \ldots &0 \\
   \vdots & \ddots & \vdots \\
  {{l_{n,1}}}& \cdots &{{l_{n,n}}}
\end{array}} \right)\]
A strictly lower triangular matrix having 0s along the diagonal as well as the upper portion, i.e., a matrix \[A = \left[ {{a_{i,j}}} \right]\]for \[{a_{i,j}} = 0\] and \[i \leqslant j\].
\[L = \left( {\begin{array}{*{20}{c}}
  0&{0 \ldots }&0 \\
   \vdots & \ddots & \vdots \\
  {{a_{n,1}}}&{{a_{n,2}} \cdots }&0
\end{array}} \right)\]

Note: A triangular matrix is also a special kind of square matrix. The transpose of an upper triangular matrix is a lower triangular matrix and vice versa. A matrix which is both symmetric and triangular is diagonal. The determinant and permanent of a triangular matrix equal the product of the diagonal entries, as can be checked by direct computation.