# Mathematics | Matrix Introduction

A matrix represents a collection of numbers arranged in an order of rows and columns. It is necessary to enclose the elements of a matrix in parentheses or brackets.

A matrix with 9 elements is shown below.

This Matrix [M] has 3 rows and 3 columns. Each element of matrix [M] can be referred to by its row and column number. For example, a_{23}=6

**Order of a Matrix :**

The order of a matrix is defined in terms of its number of rows and columns.

Order of a matrix = No. of rows ×No. of columns

Therefore Matrix [M] is a matrix of order 3 × 3.

**Transpose of a Matrix :**

The transpose [M]^{T} of an m x n matrix [M] is the n x m matrix obtained by interchanging the rows and columns of [M].

if A= [a_{ij}] mxn , then A^{T} = [b_{ij}] nxm where b_{ij} = a_{ji}

**Properties of transpose of a matrix:**

- (A
^{T})^{T}= A - (A+B)
^{T}= A^{T}+ B^{T} - (AB)
^{T}= B^{T}A^{T}

**Singular and Nonsingular Matrix:**

- Singular Matrix: A square matrix is said to be singular matrix if its determinant is zero i.e. |A|=0
- Nonsingular Matrix: A square matrix is said to be non-singular matrix if its determinant is non-zero.

**Properties of Matrix addition and multiplication:**

- A+B = B+A (Commutative)
- (A+B)+C = A+ (B+C) (Associative)
- AB ? BA (Not Commutative)
- (AB) C = A (BC) (Associative)
- A (B+C) = AB+AC (Distributive)

**Square Matrix:** A square Matrix has as many rows as it has columns. i.e. no of rows = no of columns. **Symmetric matrix:** A square matrix is said to be symmetric if the transpose of original matrix is equal to its original matrix. i.e. (A^{T}) = A. **Skew-symmetric:** A skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative.i.e. (A^{T}) = -A.

**Diagonal Matrix:**A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The term usually refers to square matrices. **Identity Matrix:**A square matrix in which all the elements of the principal diagonal are ones and all other elements are zeros.Identity matrix is denoted as I. **Orthogonal Matrix:** A matrix is said to be orthogonal if AA^{T} = A^{T}A = I **Idempotent Matrix:** A matrix is said to be idempotent if A^{2} = A **Involutary Matrix:** A matrix is said to be Involutary if A^{2} = I.

Note: Every Square Matrix can uniquely be expressed as the sum of a symmetric matrix and skew-symmetric matrix. A = 1/2 (AT + A) + 1/2 (A – AT).

**Adjoint of a square matrix:** The adjoint of a matrix A is the transpose of the cofactor matrix of A

**Properties of Adjoint:**

- A(Adj A) = (Adj A) A = |A| I
_{n} - Adj(AB) = (Adj B).(Adj A)
- |Adj A|= |A|
^{n-1} - Adj(kA) = k
^{n-1}Adj(A) - |adj(adj(A))|= |A|^(n-1)^2
- adj(adj(A))=|A|^(n-2) * A
- If A = [L,M,N] then adj(A) = [MN, LN, LM]
- adj(I) = I

Where, “n = number of rows = number of columns”

Inverse of a square matrix:

Here |A| should not be equal to zero, means matrix A should be non-singular.

**Properties of inverse:**

1. (A^{-1})^{-1} = A

2. (AB)^{-1} = B^{-1}A^{-1}

3. only a non singular square matrix can have an inverse.

**Where should we use the inverse matrix?**

If you have a set of simultaneous equations:

7x + 2y + z = 21

3y – z = 5

-3x + 4y – 2x = -1

As we know when AX = B, then X = A^{-1}B so we calculate the inverse of A and by multiplying it B, we can get the values of x, y, and z.

**Trace of a matrix:** trace of a matrix is denoted as tr(A) which is used only for square matrix and equals the sum of the diagonal elements of the matrix. Remember trace of a matrix is also equal to sum of eigen value of the matrix. For example:

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