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, 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 = [a_ij] mxn , then A^T = [b_ij] nxm where b_ij = a_ji

Properties of the transpose of a matrix:

• (A^T)^T = A
• (A+B)^T = A^T + B^T
• (AB)^T = B^TA^T

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. (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 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 AA^T = A^TA = 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 (A^T + A) + 1/2 (A – A^T). 

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

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”

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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