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# Mathematics | Matrix Introduction

• Difficulty Level : Easy
• Last Updated : 10 Jan, 2023

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, a23 = 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 = [aij] mxn , then AT = [bij] nxm where bij = aji

## Properties of the transpose of a matrix:

• (AT)T = A
• (A+B)T = AT + BT
• (AB)T = BTAT

## Singular and Nonsingular Matrix:

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

## Types of Matrices:

• 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 the original matrix is equal to its original matrix. i.e. (AT) = A.
• Skew-symmetric: A skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative.i.e. (AT) = -A.
• Diagonal Matrix: A diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The term usually refers to square diagonal 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 AAT = ATA = I
• Idempotent Matrix: A matrix is said to be idempotent if A2 = A
• Involuntary Matrix: A matrix is said to be Involuntary if A2 = I.
• Zero or Null Matrix: A matrix is said to zero or null matrix if all its elements are zero
• Upper Triangular Matrix: A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix
• Lower Triangular Matrix: A square matrix in which all the elements above the diagonal are zero is known as the lower triangular matrix

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).

## Determinant of a matrix :

The determinant of a matrix is a number associated with that square matrix. The determinant of a matrix can only be calculated for a square matrix. It is represented by |A|.

Example:

Input: 2 X 2 Matrix

Then, the determinant is -> |A| = ad – bc

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

• If A = [L,M,N] then adj(A) = [MN, LN, LM]

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

## The inverse of a square matrix:

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

### Properties of the inverse:

• (A-1)-1 = A
• (AB)-1 = B-1A-1
• 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-1B so we can calculate the inverse of A and by multiplying it by B, we can get the values of x, y, and z.

## Trace of a matrix:

The 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 the sum of the eigenvalue of the matrix. For example:

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