Upper Triangular Matrix: Definition, Types, Properties, Applications & Solved Questions

Triangular Matrix is a sort of square matrix in Linear Algebra in which the entries below and above the diagonal appear to form a triangle. There are two basic forms of triangular matrices. All elements above the main diagonal in a lower triangular matrix are zero, while all elements lying below the main diagonal are 0 in an upper triangular matrix. We may obtain the value of the determinant for any upper triangular matrix by multiplying all of the elements along the major diagonal of the matrix. This also means that if a 0 appears anywhere along an upper triangular matrix's main diagonal, the determinant will be 0.

Key Terms: Upper Triangular Matrix, Triangular Matrix, Lower Triangular Matrix, Matrices, Diagonals, Determinant, Square Matrix, Elements, Atomic Matrix, Unitriangular Matrix

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What is a Triangular Matrix?

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A triangular matrix can be defined as a special kind of square matrix in which elements below and/or above the diagonal are all zeros. There are mainly two types of triangular matrices: 

• Lower Triangular Matrix: A lower triangular matrix is a square matrix in which all the elements above its main diagonal are zero. 
• Upper Triangular Matrix: An upper triangular matrix can be defined as a square matrix in which all the elements below the main diagonal are zero.

\(A = \begin{bmatrix}2 & -1 & 3 \\[0.3em]0 & 5 & 2 \\[0.3em] 0 & 0 & -2 \\[0.3em] \end{bmatrix}, B = \begin{bmatrix}2 &0 & 0 \\[0.3em]1 & 5 &0 \\[0.3em] 1 & -1 & -2 \\[0.3em] \end{bmatrix} \)

Upper Triangular Matrix, Lower Triangular Matrix

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Matrices Detailed Video Explanation:

Upper Triangular Matrix

An n × n square matrix A = [aij] is called an upper triangular matrix if and only if aij = 0, for all i > j. In an upper triangular matrix, this means that all elements below the major diagonal of a square matrix are zero. U = [uij for I j, 0 for I > j] is a common notation for an upper triangular matrix. Here is an example of an upper triangular matrix:

\(U = \begin{bmatrix}6 &0 &8 \\[0.3em]0 & 10 &-12 \\[0.3em] 0 & 0 & 2 \\[0.3em] \end{bmatrix} \)

Upper Triangular Matrix

The right triangle matrix is also known as the upper triangular matrix, whereas the left triangular matrix is also known as the lower triangular matrix.

As a result, an upper triangular matrix is defined as one with zero entries below the main diagonal, while a lower triangular matrix is defined as one with zero entries above the main diagonal. All items below the main diagonal in the Upper triangular sparse matrix have a zero value. An upper triangular matrix is another name for this sort of sparse matrix. When you look at it graphically, you'll see that all of the elements with non-zero values are shown above the diagonal.

Read More: Matrix Multiplication

Aside from these two, there are some unique form matrices, such as the following:

1. Unitriangular Matrix is a type of unitriangular matrix.
2. Strictly Matrix with three triangles
3. Atomic Matrix with three triangles

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Types of Upper Triangular Matrix

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We look at several distinct forms of Upper triangular matrices. The following is a list of several unique triangular matrices:

• Upper Triangular Matrix: If all of the elements below the main diagonal are zero, the triangular matrix is said to be upper triangular.

Upper Triangular Matrix

• Strictly Upper Triangular Matrix: If all of the elements of the major diagonal are zero, the upper triangular matrix is said to be strictly upper triangular.

Strictly Upper Triangular Matrix

• Unit Upper Triangular Matrix: If all of the components of the main diagonal are equal to 1, the upper triangular matrix is said to be a unit upper triangular matrix.

Unit Upper Triangular Matrix

Read More: Applications of Determinants and Matrices

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Properties of Upper Triangular Matrix

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Below is a list of the most important properties of an upper triangular matrix.

• When two upper triangular matrices are added together, the result is an upper triangular matrix.
• Also, when two upper triangular matrices are multiplied, the output is an upper triangular matrix.
• If the upper triangular matrix is inversed, it will remain an upper triangular matrix.
• A lower triangular matrix will be the transpose of an upper triangular matrix, UT = L.
• If a scalar amount is multiplied by the matrix, it will remain an upper triangular matrix.

At least one element above the main diagonal must be non-zero in an upper triangular matrix. A lower triangular matrix must also include at least one non-zero member below the main diagonal.

An upper triangular matrix is usually marked by the letter U, whereas a lower triangular matrix is usually denoted by the letter L.

Read More: Inverse Matrix Formula

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Application of Upper Triangular Matrix

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The application of Upper Triangular matrices in our daily lives is far more extensive than most people realise. Every day, when we go to work, school, or university, the use or instances of matrices are in front of us. The following are some examples of matrices in action:

• Encryption: In encryption, matrices are used to jumble data for security reasons, essentially to encode or decode the data. The data can be encoded and decoded with the use of a key that is generated using matrices.
• 3D Games: Matrices are used to edit or recreate objects in 3D space, especially in games. They convert a 3d matrix to a 2d matrix to convert it into the many items required.
• Business and Economics: A matrix is used in economics and business studies to investigate business trends, share, and construct business models, among other things.
• Construction: Most structures we see are straight, however, architects occasionally construct skyscrapers with slight variations in the outside construction, such as the iconic Burj Khalifa. Matrixes are used to do this. A matrix is made up of rows and columns, as we all know. We can make such structures by changing the number of rows and columns in a matrix.
• Engineering: Engineers utilise matrices for Fourier analysis, Gauss Theorem, and finding forces in bridges, among other things. Matrix transformations are used to obtain precisely calibrated computations in chemical engineering.
• Miscellaneous: Matrices can also be found in electrical networks, planes, and spacecraft.
• Dance: Matrices are used in dance to structure complex group dances.
• Animation: Matrices can improve the precision and perfection of animations.
• Physics: Matrixes are used in the study of electrical circuits, optics, and quantum mechanics in physics. It aids us in determining battery power outputs. It is also feasible to convert electrical energy into another usable energy using matrices. As a result, we can conclude that matrices play a significant role in calculations, particularly when solving problems involving Kirchoff's voltage and current laws.
• Graphics Software: Matrixes are used in apps like Adobe Photoshop to process linear transformations and represent images.
• Geology: Matrices are also useful in seismic surveys in geology.
• Hospital: Matrixes are utilised in hospitals for medical imaging, CAT scans, and MRIs.

Read More: Invertible Matrices

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Things to Remember

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• A triangular matrix can be defined as a special kind of square matrix in the set of matrices in which elements below and/or above the diagonal are all zeros. 
• A lower triangular matrix or upper triangular matrix are two types of a triangular matrix. 
• A square matrix in which all elements above the main diagonal are zero is known as a lower triangular matrix while a square matrix whose all elements below the main diagonal are zero is called an upper triangular matrix.
• If and only if aij = 0, for all i > j, an n × n square matrix A = [aij] is said to be an upper triangular matrix. 
• The general notation of an upper triangular matrix is given as U = [uij for i ≤ j, 0 for i > j].

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