Different Types of Matrices

Matrices can be categorized based on the value of their elements, their order, the number of rows and columns, etc. Below are the various matrix types, along with their definitions and examples:

• Scalar Matrix: A scalar matrix is a matrix with all elements equal to a single scalar value. For example, a scalar matrix of order 3x3 would look like: $$\begin{bmatrix} a & a & a \ a & a & a \ a & a & a \end{bmatrix}$$

• Identity Matrix: An identity matrix is a square matrix with all elements on the main diagonal equal to 1 and all other elements equal to 0. For example, an identity matrix of order 3x3 would look like: $$\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$$

• Diagonal Matrix: A diagonal matrix is a square matrix in which all elements outside the main diagonal are equal to 0. For example, a diagonal matrix of order 3x3 would look like: $$\begin{bmatrix} a & 0 & 0 \ 0 & b & 0 \ 0 & 0 & c \end{bmatrix}$$

• Upper Triangular Matrix: An upper triangular matrix is a square matrix in which all elements below the main diagonal are equal to 0. For example, an upper triangular matrix of order 3x3 would look like: $$\begin{bmatrix} a & b & c \ 0 & d & e \ 0 & 0 & f \end{bmatrix}$$

• Lower Triangular Matrix: A lower triangular matrix is a square matrix in which all elements above the main diagonal are equal to 0. For example, a lower triangular matrix of order 3x3 would look like: $$\begin{bmatrix} a & 0 & 0 \ b & c & 0 \ d & e & f \end{bmatrix}$$

All Contents in Matrices

Introduction to Matrices

Types of Matrices

Matrix Operations

Adjoint and Inverse of a Matrix

Rank of a Matrix and Special Matrices

Solving Linear Equations using Matrix

JEE Main 2021 Maths LIVE Paper Solutions - 24th February Shift-1 (Memory-Based)

Matrix Types: Overview

The different types of matrices are:

• Identity Matrix
• Zero Matrix
• Diagonal Matrix
• Symmetric Matrix
• Skew-Symmetric Matrix
• Upper Triangular Matrix
• Lower Triangular Matrix
• Scalar Matrix
• Permutation Matrix
• Orthogonal Matrix

| — | — |

| Row Matrix | A = [a_ij]^1×n |

| Column Matrix | A = [a_ij]_m×1 |

| Zero or Null Matrix | A = [aij]mxn where, aij = 0 |

| Singleton Matrix | A = [a_ij]^{m x n} where, m = n = 1 |

| Horizontal Matrix | [aij]^{m x n} where n > m |

| Vertical Matrix | $\mathbf{A}_{i,j}^{m,n}$ where, $m > n$ |

| Square Matrix | [aij]^{m x n} where, m = n |

| Diagonal Matrix | A = [$a_{ij}$] when $i \neq j$ |

| Scalar Matrix | A = [$a_{ij}$]$m \times n$ where, $a_{ij} = \begin{cases} 0, & i \neq j \ k, & i = j \end{cases}$

Where $k$ is a constant.

| Identity (Unit) Matrix | A = [aij]m×n where, (a_{ij} = \begin{cases} 1, & \text{if } i = j \ 0, & \text{if } i \ne j \end{cases}) |

| Equal Matrices | A = [aij]mxn and B = [bij]rxs where, aij = bij, m = r, and n = s |

| Triangular Matrices | Can be either upper triangular (aij = 0, when i > j) or lower triangular (aij = 0 when i < j) |

| Singular Matrix | |A| = 0 |

| Non-Singular Matrix | |:—:|:—:| |A| ≠ 0|

| Symmetric Matrices | A = [aij] where, aij = aji |

| Skew-Symmetric Matrices | A = [aij] where aij = -aji |

| Hermitian Matrix | A = A^θ |

| Skew-Hermitian Matrix | A^θ = -A |

| Orthogonal Matrix | $A \cdot A^T = I = A^T \cdot A$ |

| Idempotent Matrix | A² = A |

| Involuntary Matrix | A2 = I, A^-1 = A |

| Nilpotent Matrix | $\exists p \in \mathbb{N}$ such that $A^p = 0$ |

#Types of Matrices: ##Explanations

Row Matrix

A matrix having only one row is called a row matrix. Thus, A = [aij]1×n is a row matrix if m = 1. It is called so because it has only one row, and the order of a row matrix will hence be 1 × n. For example, A = [1 2 4 5] is row matrix of order 1 x 4. Another example of the row matrix is P = [ -4 -21 -17 ] which is of the order 1×3.

Column Matrix

A matrix having only one column is called a column matrix. Thus, A = [aij]mx1 is a column matrix, and the order of the matrix is m × 1.

An example of a column matrix is:

$$\begin{bmatrix} 1\ 2\ 3 \end{bmatrix}$$

A = \begin{bmatrix} -1 & 2 & -4 & 5 \end{bmatrix} is a column matrix of order 4 x 1.