# Triangular Matrix:

A square matrix in which all the entries upper or below the maim diagonals elements are zero is said to be a triangular matrix. The triangular matrix must be a square matrix that means the triangular matrix has the same number of rows and columns.

There are two types of triangular matrices. the upper triangular matrix and lower triangular matrix

## Types of a Triangular Matrix:

### Upper Triangular Matrix:

A square matrix in which entries below the main diagonals are zero is called an upper triangular matrix.

#### For Example:

\begin{bmatrix}5&3&2&5\\0&7&7&9\\0&0&4&1\\0&0&0&8\end{bmatrix}

### Lower Triangular Matrix:

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

#### For Examples:

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

Notation

The capital word U is the notation of the upper triangular matrix it is also called the right triangular matrix. And the capital word L is used for the lower triangular matrix it is also called the left triangular matrix.

### Properties of an Upper Triangular Matrix:

• If we added or subtract two upper triangular matrices then the matrix that we obtain is also a upper triangular matrix
• If we multiply two upper triangular matrix the resultant matrix also upper triangular matrix
• If we take inverse of upper triangular matrix then the resultant matrix is upper triangular matrix
• Transpose  of upper triangular matrix is lower triangular matrix such that U^t = L.
• The upper triangular remain unchanged when we multiply it by any scalar quantity .

### Properties of a Lower Triangular Matrix:

• If we added or subtract two lower triangular matrices then the resultant matrix is a lower triangular matrix.
• If we multiply two lower triangular matrix the resultant matrix also lower triangular matrix
• If we take inverse of lower triangular matrix then the resultant matrix is lower triangular matrix
• Transpose  of lower triangular matrix is lower triangular matrix such that L^t</span> = U .
• The lower triangular remain unchanged when we multiply it by any scalar quantity .

Some other types of a triangular matrix

### Unit Triangular Matrix:

In any upper or lower triangular matrix if the main diagonals entries are 1 then it is called unit (upper or lower) triangular matrix. Mostly it is called normed triangular matrix

\begin{bmatrix}1&0&0&0\\6&1&0&0\\7&3&1&0\\3&5&9&1\end{bmatrix}\; lower unit triangular matrix

\begin{bmatrix}1&6&7&8\\0&1&7&3\\0&0&1&6\\0&0&0&1\end{bmatrix}\; upper unit triangular matrix

### Strictly Triangular Matrix:

In any upper or lower triangular matrix if the elements of the main diagonal are zero then it is called strictly upper or lower triangular matrix

\begin{bmatrix}0&6&7&8\\0&0&7&3\\0&0&0&6\\0&0&0&0\end{bmatrix}\; upper strictly triangular matrix.

\begin{bmatrix}0&0&0&0\\5&0&0&0\\6&1&0&0\\9&7&5&0\end{bmatrix}\; lower strictly triangular matrix.