Triangular Matrix

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.

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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:

1. Upper triangular matrix
2. Lower triangular matrix

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

1. 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).
2. Any triangular matrix of any order has a determinant equal to the product of its primary diagonal members.
3. If all of the primary diagonal members are non-zero, then a triangular matrix is invertible.
4. When two triangular matrices are multiplied, the product produced is also the triangular matrix
5. A multiplication of two upper (lower) triangular matrices yields an additional upper (lower) triangular matrix in the final matrix.
6. The resulting matrix is also an upper (lower) triangular matrix when two upper (lower) triangular matrices are joined.

Summary

Triangular matrices play a crucial role in both computational and theoretical contexts due to their simplified structure, which makes solving linear equations, computing determinants, and performing matrix factorizations more efficient. For example, algorithms such as Gaussian elimination frequently convert matrices into triangular form to simplify calculations. The unique structure and properties of triangular matrix make it suitable for use in various fields like engineering, physics, architecture, etc.

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Solved Examples Based On Triangular Matrices

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

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

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|=