A Triangular Matrix is a square matrix where the below or above diagonal elements are zero. Generally, we will have two types of triangular matrices. One is a lower triangular matrix which is a square matrix where all elements above the main diagonal are zero. The other one is the upper triangular matrix which is a square matrix where all elements below the main diagonal are zero.
Let us discuss the types of triangular matrices, and their properties and examples clearly in this article. For a better understanding of the 10th Grade Math triangular matrix concept, don’t skip any part of this article. We have given a clear explanation in every section of this article.
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What is a Triangular Matrix?
A triangular matrix is also one type of square matrix. The two types of matrices are the lower triangular matrix and an upper triangular matrix. The lower triangular matrix will have the elements above its main diagonal zero. Also, the upper triangular matrix will have elements below its main diagonal zero.
Examples of Triangular Matrix
\( A =\left[
\begin{matrix}
5&10&15 \cr
0&25&30 \cr
0&0&30 \cr
\end{matrix}
\right]
\)
It is a upper triangular matrix and \( A =\left[
\begin{matrix}
5&0&0 \cr
20&25&0 \cr
5&10&15 \cr
\end{matrix}
\right]
\)
It is a lower triangular matrix.
Types of Triangular Matrices
Different types of Triangular Matrices are given below.
- Upper Triangular Matrix: Upper Triangular Matrix is a triangular matrix where all the elements below the main diagonal are zero.
- Lower Triangular Matrix: Lower Triangular Matrix is a triangular matrix where all the elements above the main diagonal are zero.
- Strictly Triangular Matrix: Strictly Triangular Matrix is a triangular matrix where all the elements of the main diagonal are zero.
- Strictly Lower Triangular Matrix: A lower triangular matrix is known as a Strictly Lower Triangular Matrix when all the elements of its main diagonal are zero.
- Strictly Upper Triangular Matrix: A Upper triangular matrix is known as Strictly Upper Triangular Matrix when all the elements of its main diagonal are zero.
- Unit Triangular Matrix: Unit Triangular Matrix is a triangular matrix where all the elements of the main diagonal are equal to 1.
- Unit Lower Triangular Matrix: A lower triangular matrix is a Unit Lower Triangular Matrix when all the elements of the main diagonal are equal to 1.
- Unit Upper Triangular Matrix: A Upper triangular matrix is a Unit Upper Triangular Matrix when all the elements of the main diagonal are equal to 1.
Upper Triangular Matrix
In the Upper Triangular Matrix, all the elements below the diagonal are zero. If we consider n × n square matrix A = [aij], then aij = 0, for all i > j to prove it as a Upper Triangular Matrix. The notation of an upper triangular matrix is U = [uij for i ≤ j, 0 for i > j]. An example for an upper triangular matrix is
\( U =\left[
\begin{matrix}
3&4&5 \cr
0&2&4 \cr
0&0&5 \cr
\end{matrix}
\right]
\)
Lower Triangular Matrix
In the Lower Triangular Matrix, all the elements above the diagonal are zero. If we consider n × n square matrix A = [aij], then aij = 0, for all i < j to prove it as a Lower Triangular Matrix. The notation of an upper triangular matrix is L = [lij for i ≥ j, 0 for i < j]. An example of an upper triangular matrix is
\( L =\left[
\begin{matrix}
3&0&0 \cr
5&2&0 \cr
7&11&5 \cr
\end{matrix}
\right]
\)
Properties of Triangular Matrix
Know the importance of Triangular Matrix Properties and the properties of triangular matrices from the below list.
- The transpose of a triangular matrix is triangular
- The determinant of a triangular matrix is the product of all the elements present in the main diagonal.
- We will get the multiplication of two triangular matrices in a triangular matrix.
- The transpose of an upper triangular matrix is a lower triangular matrix and vice-versa.
- The inverse of a triangular matrix is triangular.
- A triangular matrix will be invertible if and only if all entries of the main diagonal are non-zero.
- The product of two upper (lower) triangular matrices is an upper (lower) triangular matrix.
- If a matrix is both upper and lower triangular, then it is known as a diagonal matrix.
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Triangular Matrix Examples
Solved Examples on a Triangular Matrix are given below with step-by-step explanations and answers. Solve every problem provided here to answer all the questions that appear in the exam.
Example 1.
Identify if the given matrix is a triangular matrix. Also, identify its type.
\( A =\left[
\begin{matrix}
5&0 \cr
4&2 \cr
\end{matrix}
\right]
\)
Solution:
Given matrix is \( A =\left[
\begin{matrix}
5&0 \cr
4&2 \cr
\end{matrix}
\right]
\)
The element above the diagonal is a12 = 0 and below the diagonal is a21 = 4. The matrix that has the elements zero above the diagonal is called the lower triangular matrix.
Therefore, the given matrix is a lower triangular matrix as the element above the main diagonal is zero.
Example 2.
Find the values of ‘m’ and ‘n’ in the given matrix A such that A is a strictly upper triangular matrix.
\( A =\left[
\begin{matrix}
3m&4 \cr
n&5 \cr
\end{matrix}
\right]
\)
Solution:
Given matrix is \( A =\left[
\begin{matrix}
3m&4 \cr
n&5 \cr
\end{matrix}
\right]
\)
For the strictly upper triangular matrix, the elements below the diagonal are zero and the elements of the main diagonal are zero.
Therefore, we must have 3m = 0; m = 0.
and n = 0.
Therefore, the values of m and n are 0.
Example 3.
Find the determinant of the matrix \( A =\left[
\begin{matrix}
5&0&0 \cr
0&a&0 \cr
1&5&b \cr
\end{matrix}
\right]
\)
Solution:
Given matrix is \( A =\left[
\begin{matrix}
5&0&0 \cr
0&a&0 \cr
1&5&b \cr
\end{matrix}
\right]
\)
All the elements above the diagonal are zero. Therefore, the given matrix is a lower triangular matrix. For a lower triangular matrix, the determinant can find by the product of its diagonal elements.
The product of diagonal elements = 5 * a * b = 5ab.
Therefore, det A = 5ab.
Example 4.
Given an Example of the Upper Triangular Matrix.
Solution:
The example of an Upper Triangular Matrix is \( U =\left[
\begin{matrix}
24&31&23 \cr
0&17&69 \cr
0&0&20 \cr
\end{matrix}
\right]
\)
FAQs on Triangular Matrix
1. What is a Triangular Matrix?
A triangular matrix is one type of square matrix whose elements above and below the diagonal appear to be in the form of a triangle. The elements either below and/or above the main diagonal of a triangular matrix are zero.
2. When is a Triangular Matrix Invertible?
When all entries of the main diagonal are non-zero then a triangular matrix (lower or upper) is invertible.
3. How to Find the Determinant of a Triangular Matrix?
Multiply the diagonal elements of a triangular matrix to determine its determinant.
4. What is the Inverse of a Lower Triangular Matrix?
The Inverse of a Lower Triangular Matrix is also a Lower Triangular Matrix.
Summary
The complete details of the triangular matrix are given along with the different types of triangular matrices. Check out all the properties of triangular matrices and solve every problem given in this article. Be the first to grab knowledge on triangular matrix. Many of the students scored good marks in the exams by referring to our articles. So, stay connected with our website to learn the best.