Upper Triangular Matrix: Overview, Questions, Preparation

A matrix is an ordered rectangular array of numbers (or functions). A 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 m x n matrix.  [ ] or ( ) encloses the arrangement. 
A typical matrix can be represented as:

The above matrix is represented by A = [aij]_mxn. The number a₁₁, a₁₂, … etc., are known as the matrix A elements, where a_ij belongs to the ith row and jth column and is called the (i, j)^th element of the matrix A = [a_ij].

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:

Triangular matrices, whether upper or lower, are very easy to solve and used in various numerical analyses.

It is represented as: 

This can also be called a right triangular matrix as the non-zero terms are concentrated on the right. It can also be defined as a square matrix that has zero entries below the main diagonal. 

Properties of Upper Triangular Matrix

1. The addition of two upper triangular matrices gives an upper triangular matrix.
2. Subtraction of two upper triangular matrices gives an upper triangular matrix.
3. The inverse of the upper triangular matrix is an upper triangular.
4. The transpose of an upper triangular matrix will always be a lower triangular matrix, UT = L.
5. Even after multiplying any scalar quantity to an upper triangular matrix, the matrix will continue to be an upper triangular matrix.

Example of Upper Triangular Matrix

While matrices and determinants carry 13 marks in the board exams, it consists of a six mark sum that is most probably coming from the matrix section. Upper triangular matrices are very popular in long sums and are a method to solve one of the more complex sums. Matrices are in general widely used in physics, geology, software, electronics, and engineering applications and such types of matrices often feature in national entrance examinations.

Calculate the following:

1.

This just 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.

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

Source: NCERT

Q: What are the other types of triangular matrices?

Q: What are the types of triangular matrices?

Q: How can we mathematically represent a triangular matrix?

A:

  Q: What is the real matrix?

Q: Define Upper and Lower Triangular Matrices.