BodhiAI

Nov. 6, 2019
**ALGEBRA OF MATRICES**

EQUALITY OF MATRICES

Two matricesA = [a_{ij}]_{m×n} and B = [b_{ij}]_{p×q} are equal if

(i) m = p

i.e. Number of rows in A equals number of rows in B.

(ii) n = q

i.e. Number of rows in A equals number of rows in B.

(iii) a_{ij} = b_{ij} ∀1<=i<=m<=m(= p) and1<=j<=m(=q)

i.e. To each and every respective element in A, ∃ a corresponding equal element in B.

ADDITION OF MATRICES

Let A=[a_{ij}]andB=[b_{ij}]be two matrices of same order say m × n. Then the matrix obtained by

adding the corresponding elements of matrices A and B is called the sum of matrices A and B.

Such that C = A + B

C_{ij}=a_{ij}+b_{ij }∀1<=i<=m; 1<=j<=n

**IMPORTANT**

Two matrices are said to be conformable for addition if and only if they are of the same order.

PROPERTIES OF ADDITION**P-1** **Matrix addition is Commutative**

i.e. A + B = B + A**P-2** **Matrix addition is Associative**

i.e. A + (B+ C) = B + (A + C) = C + (A + B)**P-3** **EXISTENCE OF IDENTITY**

Let O_{m×n} be an m × n Null Matrix and A=[a_{ij}] be also m× n matrix, then

A + O_{m×n} =A= O_{m×n} + A

**IMPORTANT**

ZERO/NULL Matrix is the identity matrix for matrix addition.

**P-4** **EXISTENCE OF INVERSE**

Let A and B be any two matrices of same order, say m × nif A + B = Om×n(identity

element for addition), then B is called the inverse of A and vice-versa. Thus, for every

matrix A=[a_{ij}]_{m×n} , there exist a matrix [– a_{ij}]_{m×n} denoted by – A, such that

A + (– A) = 0 = (– A) + A**P-5** Cancellation Laws hold good in case of addition of matrices. If A, B and C are matrices

of same order, then

A + B = A + C **LEFT CANCELLATION LAW**

⇒ B = C

Also, B + A = C + A **RIGHT CANCELLATION LAW**

⇒ B = C

**SUBTRACTION OF MATRICES**

Let A = [a_{ij}]and B = [b_{ij}]be any two matrices of same order say m×n, then their difference

A – B is a matrix D = [d ij ] of same order such that

d _{ij} = a_{ij} – b_{ij} ∀1<=i<=m; 1<=j<=n

The difference of A – B can also be expressed as A+ (– B) i.e. addition of matrix A with the

additive inverse/negative of matrix B.

**MULTIPLICATION OF A MATRIX BY A SCALAR**

Let A be any m×n matrices and λ be any scalar (real or complex) then the scalar multiple of

matrix A by λ is denoted by λA and is defined as the m×n matrix obtained by multiplying each

element of A by λ . Thus for any matrix A = [a_{ij}]_{m×n} , we have

λA=[λa_{ij}]_{m×n}

**ROPERTIES OF SCALAR MULTIPLICATION**

P-1 Scalar multiplication is distributive over addition

i.e. k (A + B) = kA + kB

P-2 If A is any m× n matrix and a and b are any two scalars, then

(a + b)A = aA + bAP

P-3 If A is any m× n matrix and kbe any scalar, then

(–k)A = – (kA) = k (–A)

**MULTIPLICATION OF TWO MATRICES**

Two matrices A and B are conformable for the product AB if the number of columns in

A(pre-multiplier) is the same as the number of rows in B (post-multiplier).

Thus, if A = [a_{ij}]_{m×n }and B = [b_{ij}]_{n×p} are two matrices of order m ×nand n ×prespectively, then their

product AB is of order m × pand is defined as