Triangular Matrix (Upper and Lower): Definition, Examples, Formula

Triangular Matrix

Edited By Komal Miglani | Updated on Jul 02, 2025 05:36 PM IST

A triangular matrix is a special kind of square matrix in math where the numbers below or above the main diagonal form a triangle shape. A matrix is just an arrangement of numbers in rows and columns. There are many kinds of matrices, including ones with just one row or one column, ones that aren't square, ones where only the diagonal has numbers and everything else is zero, ones where all numbers are zero, ones with ones on the diagonal and zeros elsewhere, and ones with this triangular pattern.

This Story also Contains

What is a Triangular Matrix?
Upper triangular matrix
Lower triangular matrix
Special Forms of Triangular Matrix
Properties of Triangular Matrices
Solved Examples Based On Triangular Matrices

Triangular Matrix

In this article, we will cover the concept of triangular matrices. This category falls under the broader category of Matrices, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

What is a Triangular Matrix?

A square matrix whose all elements above or below the principal diagonal are zero is called a Triangular. In numerical analysis, matrix equations involving triangular matrices are crucial because they are simpler to solve. If and only if all of an invertible matrix's leading main minors are non-zero, it can be expressed as the product of a lower triangular matrix L and an upper triangular matrix U using the LU decomposition matrix.

A triangular matrix is further classified into two types:

Upper triangular matrix
Lower triangular matrix

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Upper triangular matrix

A square matrix whose all elements below the principal diagonal are zero is called an upper triangular matrix.

An Upper triangular matrix is denoted by Letter ‘U’

$\begin{aligned} & \text { Or } \mathrm{A}=\left[\mathrm{a}_{\mathrm{i} \mathrm{j}}\right]_{\mathrm{m} \times \mathrm{n}} \text { is said to be upper triangular if } \mathrm{A}=\left[\mathrm{a}_{\mathrm{i} \mathrm{j}}\right]_{\mathrm{m} \times \mathrm{n}}=0 \text { when } \mathrm{i}>\mathrm{j} \text {. } \\ & \qquad\left[\begin{array}{ccccc}a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\ 0 & a_{22} & a_{23} & a_{24} & a_{25} \\ 0 & 0 & a_{33} & a_{34} & a_{35} \\ 0 & 0 & 0 & a_{44} & a_{45} \\ 0 & 0 & 0 & 0 & a_{55}\end{array}\right]\end{aligned}$

Properties of Upper triangular matrix

Numerous operations preserve upper triangularity:

Upper triangular is the product of two upper triangular matrices.
Upper triangular is the result of multiplying two upper triangular matrices.
If an upper triangular matrix exists, its inverse is also upper triangular.
Upper triangular is the result of multiplying a scalar by an upper triangular matrix.

Lower triangular matrix

A square matrix whose all elements above the principal diagonal are zero is called a lower triangular matrix.

The Lower triangular matrix is denoted by ‘L’

Or $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{m} \times \mathrm{n}}$ is said to be upper triangular if $\mathrm{A}=\left[\mathrm{a}_{\mathrm{i} j}\right]_{\mathrm{m} \times \mathrm{n}}=0$ when $\mathrm{i}<\mathrm{j}$.

Example,
$
\text {}\left[\begin{array}{ccccc}
a_{11} & 0 & 0 & 0 & 0 \\
a_{21} & a_{22} & 0 & 0 & 0 \\
a_{31} & a_{32} & a_{33} & 0 & 0 \\
a_{41} & a_{42} & a_{43} & a_{44} & 0 \\
a_{51} & a_{52} & a_{53} & a_{54} & a_{55}
\end{array}\right]
$

Properties of Lower triangular matrix

Numerous operations preserve lower triangularity:

Lower triangular is the product of two upper triangular matrices.
Lower triangular is the result of multiplying two upper triangular matrices.
If a Lower triangular matrix exists, its inverse is also upper triangular.
Lower triangular is the result of multiplying a scalar by an upper triangular matrix.

Special Forms of Triangular Matrix

Unit triangular matrix

A (upper or lower) triangular matrix is referred to as (upper or lower) unit triangular if all of the elements on the major diagonal are 1.

Unit (upper or lower) triangular and, extremely infrequently, normed (upper or lower) triangular are other terms for these matrices. A normed triangular matrix is unrelated to the concept of matrix norm, and a unit triangular matrix is not the same as a unit matrix.

Strictly triangular matrix

A matrix is referred to as strictly (upper or lower) triangular if every entry on the main diagonal of the matrix is likewise 0. The Cayley-Hamilton theorem states that all finite strictly triangular matrices are nilpotent of the index at most n.

Properties of Triangular Matrices

A lower triangular matrix is a transpose of an upper triangular matrix (UT = L), while an upper triangular matrix is a transpose of a lower triangular matrix (LT = U).
Any triangular matrix of any order has a determinant equal to the product of its primary diagonal members.
If all of the primary diagonal members are non-zero, then a triangular matrix is invertible.
When two triangular matrices are multiplied, the product produced is also the triangular matrix
A multiplication of two upper (lower) triangular matrices yields an additional upper (lower) triangular matrix in the final matrix.
The resulting matrix is also an upper (lower) triangular matrix when two upper (lower) triangular matrices are joined.

Solved Examples Based On Triangular Matrices

Example 1: If A is a Lower triangular matrix with the definition

$\begin{aligned} a_{i j} & =\{i-j ; \text { when } i>j \\ & =\{i+j ; \text { when } i=j \\ & =\{0 ; \text { when } i<j\end{aligned}$

and the order of A is 3 x 3. Then the value of |A| =

1)24

2)12

3) 48

4)50

Solution:

Lower Triangular Matrix -A square matrix in which all the elements above the principal diagonal are Zero

$a_{i j}=0, i<j$

$\begin{aligned} & a_{11}=1+1=2 ; a_{22}=2+2=4 ; a_{33}=3+3=6 \\ & |A|=a_1 \times a_{22} \times a_{33}=2 \times 4 \times 6=48\end{aligned}$

Hence the value of |A| =48

Example 2: If A is a strictly triangular matrix of order 3 x 3 and $B=\operatorname{diag}\left[\begin{array}{lll}3 & 5 & 2\end{array}\right]$ ; Then |AB|=

1)30

2)5

3) 0

4)28

Solution:

Strictly triangular matrix: $a_{i i}=0$ for $1 \leq i \leq n$

Where $
A=\left[a_{i j}\right]_{n \times n}
$

Since diagonal elements of A are 0 and B is a diagonal matrix, If we multiply them we get a matrix with determinant 0 since the first column and last row have all elements = 0

Hence the value of |AB| =0

Example 3: If $A=\left[\begin{array}{ll}3 & 2 \\ 0 & 5\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right]$ ; then Which of the following is a triangular matrix?

1)A + B

2)A - B

3)AB

4)none of these

Solution:

Triangular Matrix -Upper Triangular or Lower Triangular matrix

$A B=\left[\begin{array}{ll}3 & 2 \\ 0 & 5\end{array}\right]\left[\begin{array}{ll}1 & 0 \\ 2 & 3\end{array}\right]=\left[\begin{array}{cc}7 & 6 \\ 10 & 15\end{array}\right]$

So none of these is a triangular matrix

Hence the correct option is 4) None of these

Example 4: If A is an upper triangular matrix of order 3 x 3 then which of the following is TRUE?

$
\text { 1) } a_{i j}=0
$
$\text { 2) } a_{i j}=0 \forall i>j$
$
\text { 3) } a_{i j}=0 \forall i<j
$

4) None of these

Solution:

adj with i > j denotes the elements which are below the diagonal. Example $a_{21}$ is below the diagonal.

For upper triangular matrices, $a_{i j}=0 \forall i>j$ i.e. elements below the diagonal are 0.

Hence, the answer is option 4.

Frequently Asked Questions (FAQs)

1. What is Triangular Matrix?

A square matrix whose all elements above or below the principal diagonal are zero is called a Triangular matrix. We can use Triangular matrices for solving Linear equations and systems of linear equations. They can be applied to the computation of a matrix's determinant, a measure of the relationship between the matrix and its constituent parts.

2. What is the Upper Triangular Matrix?

A square matrix whose all elements below the principal diagonal are zero is called an upper triangular matrix. It is denoted by the letter ‘U’.

3. What is the Lower Triangular Matrix?

square matrix whose all elements above the principal diagonal are zero is called a lower triangular matrix. It is denoted by the letter ‘L’.

4. How to find the determinant of upper triangular matrices?

The upper triangular matrix of any order has a determinant equal to the product of its primary diagonal members.

5. What is the Unit triangular Matrix?

A (upper or lower) triangular matrix is referred to as (upper or lower) unit triangular if all of the elements on the major diagonal are 1. Unit (upper or lower) triangular and, extremely infrequently, normed (upper or lower) triangular are other terms for these matrices. A normed triangular matrix is unrelated to the concept of matrix norm, and a unit triangular matrix is not the same as a unit matrix.

6. How does an upper triangular matrix differ from a lower triangular matrix?

An upper triangular matrix has all elements below the main diagonal equal to zero, while a lower triangular matrix has all elements above the main diagonal equal to zero. The main diagonal can contain non-zero elements in both cases.

7. How do you solve a system of linear equations represented by a triangular matrix?

Systems represented by triangular matrices can be solved easily using back-substitution (for upper triangular) or forward-substitution (for lower triangular). This process is much simpler than solving general systems of linear equations.

8. Can you add two triangular matrices of different types?

Yes, you can add an upper triangular matrix to a lower triangular matrix. The result, however, will generally not be a triangular matrix unless one of the matrices is the zero matrix.

9. How do triangular matrices appear in the context of matrix approximation?

In matrix approximation problems, such as low-rank approximations, triangular matrices often play a role. For example, the truncated QR decomposition provides an upper triangular matrix that can be used for efficient low-rank approximations.

10. What is the role of triangular matrices in Cholesky decomposition?

Cholesky decomposition expresses a Hermitian positive-definite matrix as the product of a lower triangular matrix and its conjugate transpose. This decomposition is useful in many numerical algorithms, including solving systems of linear equations.

11. What is the determinant of a triangular matrix?

The determinant of a triangular matrix is the product of its diagonal elements. This is true for both upper and lower triangular matrices.

12. How do you multiply two upper triangular matrices?

When multiplying two upper triangular matrices, the result is always an upper triangular matrix. The multiplication follows the usual matrix multiplication rules, but many calculations are simplified due to the zero entries.

13. Is the inverse of a triangular matrix always triangular?

Yes, if a triangular matrix is invertible, its inverse is also triangular. An upper triangular matrix will have an upper triangular inverse, and a lower triangular matrix will have a lower triangular inverse.

14. Can a triangular matrix be singular?

Yes, a triangular matrix can be singular. It is singular if and only if at least one of the elements on its main diagonal is zero, as this would make the determinant zero.

15. What is the relationship between triangular matrices and matrix factorization?

Triangular matrices play a crucial role in various matrix factorization techniques. For example, LU decomposition expresses a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U).

16. What is a strictly triangular matrix?

A strictly triangular matrix is a triangular matrix where all elements on the main diagonal are also zero. This means all non-zero elements are either strictly above (strictly upper triangular) or strictly below (strictly lower triangular) the main diagonal.

17. How do triangular matrices relate to matrix Jordan canonical form?

The Jordan canonical form of a matrix is always upper triangular. This form is particularly useful for understanding the structure of a matrix in terms of its eigenvalues and generalized eigenvectors.

18. Can a triangular matrix be symmetric?

A triangular matrix can only be symmetric if it is diagonal (i.e., all off-diagonal elements are zero). This is because symmetry requires the matrix to be equal to its transpose, which is only possible for a triangular matrix if it's diagonal.

19. How does the concept of triangular matrices extend to block matrices?

In block matrices, the concept of triangular matrices extends to block triangular matrices. These are matrices where the blocks below (or above) the diagonal blocks are zero matrices, analogous to lower (or upper) triangular matrices.

20. How do you calculate the power of a triangular matrix?

When raising a triangular matrix to a power, the result remains triangular. For an upper triangular matrix A, A^n is obtained by raising each diagonal element to the nth power and adjusting the above-diagonal elements accordingly.

21. What is the connection between triangular matrices and Gaussian elimination?

Gaussian elimination, when applied to a square matrix, essentially transforms it into an upper triangular matrix. This process is fundamental in solving systems of linear equations and finding matrix inverses.

22. How do triangular matrices relate to Schur decomposition?

The Schur decomposition states that every square matrix is unitarily similar to an upper triangular matrix. This decomposition is a generalization of diagonalization and is always possible, even when diagonalization is not.

23. What is the role of triangular matrices in the QR algorithm for eigenvalue computation?

The QR algorithm, used for computing eigenvalues, involves iteratively factoring a matrix into a product of an orthogonal matrix Q and an upper triangular matrix R. The algorithm converges to an upper triangular matrix whose diagonal entries are the eigenvalues.

24. What is the relationship between triangular matrices and matrix normal forms?

Triangular matrices are closely related to several matrix normal forms. For instance, the Schur form is upper triangular, and the Jordan normal form is upper triangular with a specific structure on the superdiagonal.

25. How do triangular matrices appear in the study of matrix Lie groups?

The group of invertible upper (or lower) triangular matrices forms an important example of a matrix Lie group. These groups, known as Borel subgroups, play a crucial role in the theory of Lie groups and algebraic groups.

26. What is a triangular matrix?

A triangular matrix is a square matrix where all entries either above or below the main diagonal are zero. There are two types: upper triangular (zeros below the diagonal) and lower triangular (zeros above the diagonal).

27. Can a triangular matrix be non-square?

No, a triangular matrix must be square. The concept of "above" or "below" the main diagonal only makes sense in a square matrix where the number of rows equals the number of columns.

28. Can a triangular matrix have complex entries?

Yes, a triangular matrix can have complex entries. The properties of triangular matrices generally hold whether the entries are real or complex numbers.

29. What is the rank of a triangular matrix?

The rank of a triangular matrix is equal to the number of non-zero elements on its main diagonal. This is because the rank is determined by the number of linearly independent rows or columns.

30. What is the connection between triangular matrices and matrix similarity?

Two matrices are similar if one can be obtained from the other by a similarity transformation. Every square matrix is similar to an upper triangular matrix (its Schur form), which is fundamental in understanding matrix structure and properties.

31. What is the significance of triangular matrices in linear differential equations?

When solving systems of linear differential equations, if the coefficient matrix is triangular, the system can be solved by a simple process of back-substitution or forward-substitution, greatly simplifying the solution process.

32. How do triangular matrices relate to matrix polynomials?

When evaluating matrix polynomials, if the matrix is triangular, the resulting polynomial is also triangular. This property can significantly simplify calculations involving matrix polynomials.

33. What is the role of triangular matrices in the study of matrix pencils?

In the theory of matrix pencils (A - λB), triangular matrices play a role in canonical forms and decompositions. For example, the Weierstrass canonical form involves triangular matrices when dealing with regular matrix pencils.

34. What is the significance of triangular matrices in the study of matrix equations?

Many matrix equations, such as AX = B or AX - XB = C, become much easier to solve when one or more of the matrices involved are triangular. This is due to the simplified structure and properties of triangular matrices.

35. How do triangular matrices relate to the concept of matrix inertia?

The inertia of a Hermitian matrix (the number of positive, negative, and zero eigenvalues) can be determined from an LDL^T decomposition, where L is lower triangular and D is diagonal. This connection highlights the importance of triangular matrices in studying matrix definiteness.

36. What is the role of triangular matrices in the study of matrix stability?

In stability analysis, particularly for linear systems, if the system matrix can be transformed to a triangular form, the stability analysis becomes much simpler as the eigenvalues are directly visible on the diagonal.

37. What is the significance of triangular matrices in the study of matrix functions?

When computing matrix functions (like the matrix exponential or matrix logarithm), if the matrix is triangular, the computation can often be simplified. This is because many matrix functions preserve the triangular structure.

38. What is the role of triangular matrices in the study of matrix norms?

Certain matrix norms, such as the Frobenius norm, can be computed more efficiently for triangular matrices due to their structure. Additionally, the relationship between different matrix norms can sometimes be more easily understood or bounded for triangular matrices.

39. What is the trace of a triangular matrix?

The trace of a triangular matrix (sum of diagonal elements) is the same as for any square matrix. It's the sum of the elements on the main diagonal, which are the only non-zero diagonal elements in a triangular matrix.

40. How does the eigenvalue calculation simplify for triangular matrices?

For triangular matrices, the eigenvalues are simply the elements on the main diagonal. This greatly simplifies the characteristic equation and eigenvalue calculation process.

41. What is the nullity of a triangular matrix?

The nullity of a triangular matrix is the number of zero elements on its main diagonal. This is because the nullity is the dimension of the null space, which is complementary to the rank of the matrix.

42. What is the relationship between triangular matrices and matrix diagonalization?

A matrix is diagonalizable if and only if it's similar to a diagonal matrix. Triangular matrices are not always diagonalizable, but when they are, the diagonal matrix in the diagonalization is composed of the triangular matrix's diagonal elements.

43. What is the connection between triangular matrices and matrix nilpotency?

A nilpotent matrix is one that becomes the zero matrix when raised to some power. Strictly triangular matrices are always nilpotent, with the nilpotency index at most equal to the size of the matrix.

44. What is the importance of triangular matrices in numerical linear algebra?

Triangular matrices are crucial in numerical linear algebra because many algorithms, such as LU decomposition and QR factorization, involve transforming matrices into triangular form to simplify computations and improve numerical stability.

45. How does the condition number of a triangular matrix compare to general matrices?

The condition number of a triangular matrix is often lower than that of a general matrix, making triangular systems more numerically stable to solve. This is one reason why decomposing matrices into triangular factors is useful in numerical methods.

46. What is the relationship between triangular matrices and matrix exponentiation?

The exponential of a triangular matrix is also triangular. This property is useful in solving certain differential equations and in computing matrix functions.

47. How do you compute the determinant of a block triangular matrix?

The determinant of a block triangular matrix is the product of the determinants of its diagonal blocks. This is an extension of the property that the determinant of a triangular matrix is the product of its diagonal elements.

48. How do triangular matrices appear in the context of Markov chains?

In the study of Markov chains, transition matrices that are triangular represent systems where states can only transition in one direction (either to higher-numbered or lower-numbered states). This structure simplifies many calculations and analyses.

49. How do triangular matrices appear in the context of Lie algebras?

In the study of Lie algebras, the algebra of upper (or lower) triangular matrices forms an important example of a solvable Lie algebra. This connection is fundamental in the structure theory of Lie algebras.

50. How do triangular matrices relate to matrix decomposition techniques?

Many matrix decomposition techniques, such as LU, QR, and Cholesky decompositions, involve expressing a matrix as a product of triangular matrices or a triangular matrix and another special type of matrix. These decompositions are fundamental in numerical linear algebra.

51. What is the significance of triangular matrices in solving Lyapunov equations?

When solving Lyapunov equations (AX + XA^T = Q), if A is triangular, the equation can be solved much more efficiently. This is particularly useful in control theory and stability analysis of dynamical systems.

52. What is the connection between triangular matrices and Toeplitz matrices?

A triangular Toeplitz matrix is a special case where the matrix is both triangular and Toeplitz (constant along diagonals). These matrices have special properties and appear in various applications, including signal processing.

53. How do triangular matrices relate to the concept of matrix factorization in machine learning?

In machine learning, matrix factorization techniques often involve decomposing a matrix into the product of simpler matrices. Triangular matrix factorizations, like LU decomposition, can be used to simplify computations in certain machine learning algorithms.

54. How do triangular matrices appear in the context of matrix completion problems?

In matrix completion problems, where the goal is to fill in missing entries of a partially observed matrix, the problem can sometimes be simplified if the matrix is known to be triangular. This structure provides additional constraints that can aid in the completion process.

55. How do triangular matrices relate to the concept of matrix preconditioning?

In numerical linear algebra, triangular matrices are often used as preconditioners to improve the convergence of iterative methods for solving linear systems. Incomplete LU factorization, which produces triangular factors, is a common preconditioning technique.