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 transpose of a matrix:

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

Singular and Nonsingular Matrix:

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

Properties of Matrix addition and multiplication:

1. A+B = B+A (Commutative)
2. (A+B)+C = A+ (B+C) (Associative)
3. AB ≠ BA (Not Commutative)
4. (AB) C = A (BC) (Associative)
5. 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^TA = I
Idemponent Matrix: A matrix is said to be idemponent if A² = A
Involutary Matrix: A matrix is said to be Involutary if A² = 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:

Properties of Adjoint:

1. A(Adj A) = (Adj A) A = |A| I_n
2. Adj(AB) = (Adj B).(Adj A)
3. |Adj A|= |A|^n-1
4. Adj(kA) = k^n-1 Adj(A)

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^-1A^-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^-1B 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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