What is Upper Triangular Matrix: Examples and Questions

A matrix is an ordered rectangular array of numbers (or functions). The rectangular arrangement of m x n numbers (real or complex) in m rows and n columns is called a matrix having an order of m by n. It is written as an m x n matrix.  [ ] or ( ) encloses the arrangement. 
A typical matrix can be represented as:

Upper Triangular Matrix

A square matrix where all the elements above the diagonal are non-zero and below it are zero is called an upper triangular matrix. It can be represented as:

It is represented as: 

NEET and CUET exams ask questions related to the properties of upper triangular matrix. The following points highlight the properties of an upper triangular matrix:

• Eigenvalues of an upper triangular matrix are the diagonal elements. This simplifies the process of finding eigenvalues.
• An upper triangular matrix is invertible if and only if all diagonal elements are non-zero.
• If there is an inverse of upper triangular matrix, then that inverse will also be an upper triangle.
• Sum and product of 2 upper triangular matrices also result in upper triangular.
• Determinant of upper triangular matrix is product of diagonal elements.
• Rank of upper triangular matrix is (at minimum) as large as number of non-zero diagonal elements.
• Since there are many zero elements in the upper triangular matrix, different calculations are simplified.

Let us take a look at some examples of Upper triangular matrix that are often asked in exams like IISER and GATE as questions:

1. Calculate the following:

This exemplifies the upper triangular matrix’s addition property, where the sum of two upper triangular matrices gives an upper triangular matrix itself.

2. 

The product of a scalar quantity with an upper triangular matrix yields an upper triangular matrix itself, as shown in this problem.

3. Let us take a look at another example of upper triangular matrix/

This shows that the sum of an upper triangular matrix with a normal matrix does not give an upper triangular matrix.

There are a few differences between an upper triangular matrix and a lower triangular matrix. Let us discuss them in a tabular format:

Parameter	Upper Triangular Matrix	Lower Triangular Matrix
Definition	A type of square matrix where every element below the main diagonal is zero.	A type of square matrix where every element above the main diagonal is zero.
Usage	Used in Gaussian elimination and back substitution.	Used in forward substitution.

There are a number of similarities between an Upper Triangular Matrix and a Lower Triangular Matrix. Let us take a look at them:

Parameter	Upper Triangular Matrix	Lower Triangular Matrix
Non-zero Elements	Elements above and on the main diagonal may be non-zero.	Elements below and on main diagonal may be non-zero.
Determinant	Product of diagonal elements.	Product of diagonal elements.
Inverse	The inverse of an upper triangular matrix will also be an upper triangular matrix.	The inverse of a lower triangular matrix will also be lower triangular matrix.
Sum and Product	Sum and product of 2 upper triangular matrices result in upper triangular.	The sum and product of two lower triangular matrices are also lower triangular.