There are two types of triangular matrices.

**1.** Upper Triangular Matrix: A square matrix (a_{ij})

is said to be an upper triangular matrix if all the elements below the principal

diagonal are zero (0). That is, [a_{ij}]_{m}_{× n} is an

upper triangular matrix if (i) m = n and (ii) a_{ij} = 0 for i > j.

Examples of an Upper Triangular Matrix are:

(i) (begin{bmatrix} 5 & 2 & 8\ 0 & 3 &

10\ 0 & 0 & 8 end{bmatrix})

(ii) (begin{bmatrix} -1 & 7 & 3\ 0 & 6 & 1\ 0 & 0 & 5 end{bmatrix})

(iii) (begin{bmatrix} 3 & 0 & 3\ 0 & 7 & -1\ 0 & 0 & 2 end{bmatrix})

2. Lower Triangular Matrix: A square matrix (a_{ij})

is said to be a lower triangular matrix if all the elements above the principal

diagonal are zero (0). That is, [a_{ij}]_{m}_{× n} is a

lower triangular matrix if (i) m = n and (ii) a_{ij} = 0 for i < j.

Examples of a Lower Triangular Matrix are:

(i) (begin{bmatrix} 7 & 0 & 0\ 3 & 9 &

0\ 1 & 2 & 1 end{bmatrix})

(ii) (begin{bmatrix} 1 & 0 & 0\ -5 & 1 &

0\ 3 & 7 & 1 end{bmatrix})

(iii) (begin{bmatrix} 9 & 0 & 0\ 1 & 3 &

0\ 2 & 5 & -4 end{bmatrix})

**Definition of Triangular
Matrix:**

A square matrix is said to be a triangular matrix if it is

either upper triangular or lower triangular.

For example:

(i) (begin{bmatrix} 2 & 3 & 1\ 0 & 1 &

3\ 0 & 0 & 4 end{bmatrix})

(ii) (begin{bmatrix} 1 & 0 & 0\ 2 & 3 &

0\ 4 & 1 & 2 end{bmatrix})

(iii) (begin{bmatrix} 0 & 0 & 0\ 3 & 0 &

0\ 2 & 1 & 0 end{bmatrix})

(iv) (begin{bmatrix} 0 & 1 & 2\ 0 & 0 &

3\ 0 & 0 & 0 end{bmatrix})

A diagonal matrix is both upper triangular and lower

triangular.

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