A square matrix in which all the entries above the main diagonal are zero, is called a lower triangular matrix.

Let’s understand the meanings of the words “lower” and “triangular” firstly before learning the concept of the lower triangular matrix and its internal structure.

- The word “Lower” means situated below another part.
- The word “Triangular” means shaped like a triangle.

The entries above the main diagonal are zero in a square matrix in a special case. The elements on the leading diagonal and the entries below the primary diagonal form a triangle shape. Therefore, the square matrix is called a lower triangular matrix.

$L \,=\, \begin{bmatrix} e_{11} & 0 & 0 & \cdots & 0\\ e_{21} & e_{22} & 0 & \cdots & 0\\ e_{31} & e_{32} & e_{33} & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ e_{n1} & e_{n2} & e_{n3} & \cdots & e_{nn} \end{bmatrix}$

Basically, the matrix $L$ is a square matrix of order $n$. The elements $e_{11}$, $e_{22}$, $e_{33}$, $e_{44}$ $\cdots$ $e_{nn}$ are the entries on the major diagonal. The elements on the leading diagonal and the entries below the principal diagonal form a triangle shape but the elements above the primary diagonal are zeros. Hence, the matrix $L$ is called a lower triangular matrix.

$L \,=\, {\begin{bmatrix} e_{\displaystyle ij} \end{bmatrix}}_{\displaystyle n \times n}$

There is a mathematical condition in the case of a lower triangular matrix. The entries are zero if $i < j$ and they are elements above the leading diagonal of the matrix. The remaining entries are non-zeros and form a triangle shape.

It means $e_{ij} = 0$, if $i < j$.

Observe the below three matrices to know the concept of lower triangular matrix.

$A \,=\, \begin{bmatrix} 1 & 0\\ 6 & -5 \end{bmatrix}$

The matrix $A$ is a second order square matrix.

- The elements $1$ and $-5$ are entries on the major diagonal.
- The entry below the main diagonal is $6$ but the entry above the principal diagonal is $0$.
- The entries on the main diagonal $1$ and $-5$, and the entry below the primary diagonal $6$ form a triangle shape.

Hence, the matrix $A$ is called a lower triangular matrix.

$B \,=\, \begin{bmatrix} 9 & 0 & 0\\ 4 & 1 & 0\\ 7 & 2 & -3 \end{bmatrix}$

The matrix $B$ is a third order square matrix.

- The entries $9$, $1$ and $-3$ are elements on the principal diagonal.
- The elements below the leading diagonal are $4$, $7$ and $2$ but the entries above the primary diagonal are $0$.
- The diagonal entries $9$, $1$ and $-3$, and the elements below the main diagonal form a triangle shape.

So, the matrix $B$ is called a lower triangular matrix.

$C = \begin{bmatrix} 5 & 0 & 0 & 0\\ -1 & 6 & 0 & 0\\ 8 & 3 & 9 & 0\\ -2 & -6 & -7 & 2\\ \end{bmatrix}$

The matrix $C$ is a fourth order square matrix.

- The entries $5$, $6$ and $9$ and $2$ are diagonal elements on the main diagonal.
- The elements below the primary diagonal are $-1$, $8$, $3$, $-2$, $-6$ and $-7$ but the entries above the major diagonal are $0$.
- The diagonal elements and the entries below the leading diagonal form a triangle shape.

Therefore, the matrix $C$ is called a lower triangular matrix.

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