Inverses Of Triangular Matrices

Table of Contents

Triangular matrices are diagonal matrix plus some elements on the upper side or lower side of the main diagonal. Here we discuss about types, interesting properties of triangular matrix. Later we will find ways to determine the invertibility and how to find inverse of a triangular matrix with examples.

What Are Triangular Matrices ?

If a square matrix has all zeros below its main diagonal entries then it is called upper triangular matrix and if the square matrix has all zeros above the main diagonal entries then it is called lower triangular matrix. Therefore, a triangular matrix has either all zero entries above or below main diagonal.

Examples of Upper Triangular Matrix

\begin{aligned} &A = \begin{bmatrix}a_{11} & a_{12}\\0 & a_{22}\end{bmatrix}\\\\ &B = \begin{bmatrix}a_{11} & a_{12}& a_{13}\\0 & a_{22}& a_{23}\\0 & 0 & a_{33}\end{bmatrix}\\\\ &C = \begin{bmatrix}2 & 1 & 9 & 3\\0 & -1 & 8 &-2\\0 & 0 & -7 & 5\\0& 0 & 0 & 4\end{bmatrix} \end{aligned}

Examples of Lower Triangular Matrix

\begin{aligned} &P = \begin{bmatrix}a_{11} & 0\\a_{12} & a_{22}\end{bmatrix}\\\\ &Q = \begin{bmatrix}a_{11} & 0& 0\\a_{21} & a_{22}& 0\\a_{31} & a_{32} & a_{33}\end{bmatrix}\\\\ &R = \begin{bmatrix}-1 & 0 & 0 & 0\\5 & 1 & 0 &0\\-3 & 1 & -7 & 0\\8& 2 & -6 & 2\end{bmatrix} \end{aligned}

Rules Regarding Triangular Matrices

Here are some basic rules regarding the upper or lower triangular matrices.

If a square matrix $A= [a_{ij}]$ is upper triangular matrix then $a_{ij} = 0$ and $i < j$ .
If a square matrix $A= [a_{ij}]$ is lower triangular matrix then $a_{ij} = 0$ and $i > j$ .
If a square matrix $A= [a_{ij}]$ is upper triangular matrix then $i^{th}$ row has and starts with $i - 1$ zeros.
If a square matrix $A= [a_{ij}]$ is lower triangular matrix then $i^{th}$ row has and starts with $j - 1$ zeros.

Strict Triangular Matrix

A strict triangular matrix is a square matrix which has all entries zero below or above the main diagonal including the main diagonal.

Strict Upper Triangular Matrix

If a strict triangular matrix has all zero entries below the main diagonal including the main diagonal then it is called strict upper triangular matrix.

Example of Strict Upper Triangular Matrix

\begin{aligned} &A = \begin{bmatrix}0 & b\\0 & 0\end{bmatrix}\\\\ &B = \begin{bmatrix}0 & 4 & 2\\0 & 0 & 1\\0 & 0 & 0\end{bmatrix} \end{aligned}

Examples of Strict Lower Triangular Matrix

\begin{aligned} &C = \begin{bmatrix}0 & 0\\c & 0\end{bmatrix}\\\\ &D = \begin{bmatrix}0 & 0 & 0\\7 & 0 & 0\\3 & 9 & 0\end{bmatrix} \end{aligned}

The basic rules for constructing strict upper or strict lower triangular matrix are

If a square matrix $A = [a_{ij}]$ is strict upper triangular matrix when $a_{ij} = 0$ and $i \leq j$ .
If a square matrix $A = [a_{ij}]$ is strict lower triangular matrix when $a_{ij} = 0$ and $i \geq j$ .

Properties Of Triangular Matrix

There are some interesting properties of triangular matrices which we will explore each type of triangular matrices with examples.

Properties of Upper Triangular Matrices

The upper triangular matrix is in row-echelon form. Here are the properties of upper triangular matrices.

Adding two upper triangular matrices of same order will result in a upper triangular matrix of same order.
Product of two defined ( eligible for multiplication) upper triangular matrices is upper triangular matrix.
Transpose of upper triangular matrix is lower triangular matrix.
Inverse of upper triangular matrix is also upper triangular matrix.

Let us verify each claim with examples.

Addition of Two Upper Triangular Matrices

Let $A$ and $B$ be two upper triangular matrices of order $3 \times 3$ .