A matrix of order m × n is an array of m × n numbers (real or complex) in the form of m horizontal lines (called rows) and n vertical lines (called columns), enclosed by [ ] or (). In this article, we will learn the meaning of matrices, types of matrices, important formulas, etc.

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

Introduction to Matrices

An $m \times n$ matrix is usually written as:

(\begin{array}{l}A=\left[ \begin{matrix} {{a}_{11}} & {{a}_{12}} & \ldots & {{a}_{1n}} \ {{a}_{21}} & {{a}_{22}} & \ldots & {{a}_{2n}} \ \vdots & \vdots & \vdots & \vdots \ {{a}_{m1}} & {{a}_{m2}} & \ldots & {{a}_{mn}} \ \end{matrix} \right]\end{array} )

The matrix A = [aij] mxn is represented above. The elements a11, a12, etc. are referred to as the elements of A, with aij belonging to the i^th row and j^th column, and is known as the (i, j)^th element of A = [aij].

Important Formulas for Matrices

If $A, B$ are square matrices of order $n$, and $I_n$ is a corresponding unit matrix, then

(a) |A| = A if A is an adjective

(b) | adj A^T | = | A^-1 |^T (Thus A^T (adj A)^T is always a scalar matrix)

(c) Adj (adj.A) = |A|^n-2A

(\begin{array}{l}(e)\ |adj,(adj.A)| = |A^{{(n-1)}^{2}}|\end{array})

(f) (Adj A) (Adj B) = (Adj B) (Adj A)

(g)^m = (adj A)^m,

((h)\ adj (kA) = k^{n-1} \cdot adj(A), \quad k \in \mathbb{R})

((i)\ \text{adj}(I_n) = I_n)

(j) Adj 0 = 0

(k) A is symmetric ⇒ A is also symmetric

A is diagonal ⇒ A is also diagonal

(m) A is triangular ⇒ A is also triangular

(n) The singular of A is 0.

Types of Matrices

(i) Symmetric Matrix: A square matrix A = [aij] is said to be symmetric if aij = aji for all i and j.

(ii) Skew-Symmetric Matrix: when a_ij = -a_ji

(iii) Hermitian and Skew-Hermitian Matrix: A = ${{A}^{\theta }}$ (Aθ represents conjugate transpose)

(\begin{array}{l}{A}^{\theta }=-A^{\theta }\end{array} ) (skew-Hermitian matrix)