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## Properties and Examples of Upper Triangular Matrix

Upper Triangular Matrix – Introduction: An upper triangular matrix is a square matrix that has all of its elements below the main diagonal equal to zero. The elements above the main diagonal are all positive integers. Upper triangular matrices are often used in solving systems of linear equations.

Some properties of upper triangular matrices include:

- The determinant of an upper triangular matrix is always positive.
- inverse of an upper triangular matrix is also upper triangular.
- The product of two upper triangular matrices is also upper triangular.
- Some examples of upper triangular matrices include:
- The matrix A = [1, 2, 0; 3, 4, 0; 5, 6, 0] is an upper triangular matrix.
- matrix B = [1, 2, 3; 4, 5, 6; 7, 8, 9] is also an upper triangular matrix.
- The matrix C = [1, 0, 1; 0, 1, 2; 1, 0, 3] is not an upper triangular matrix, because the element in the second row and first column is not equal to zero.

## Triangular Matrices

A triangular matrix is a square matrix that has all of its entries below the main diagonal equal to zero. A triangular matrix is also called an “upper triangular matrix” if all of its entries above the main diagonal are also equal to zero.

## Properties of Upper Triangular Matrices

An upper triangular matrix is a square matrix in which all of the elements below the main diagonal are zero. This makes the upper triangular matrix easy to solve, as all of the nonzero elements are located in the main diagonal and the upper left corner. Upper triangular matrices have several properties that make them useful in mathematics and also engineering.

First, an upper triangular matrix is always positive definite. This means that the matrix has a positive determinant, which is a measure of the strength of the matrix. Second, an upper triangular matrix is always symmetric. This means that the matrix is invariant under reflection across the main diagonal. Finally, an upper triangular matrix is always positive semidefinite. This means that the matrix has a positive semidefinite determinant, which is a weaker form of the positive definite determinant.What are the Applications of Matrices?

Matrices used in various applications including the following:

- In physics, matrices used to represent the state of a system of particles and to solve physical problems.
- Engineering, matrices used to solve problems involving vibrations, forces, and also structural analysis.
- In mathematics, matrices used to solve problems in linear algebra and to carry out mathematical operations.