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BodhiAI
Nov. 6, 2019

Algebra of Matrices, Minors & Cofactors:

ALGEBRA OF MATRICES
EQUALITY OF MATRICES
Two matricesA = [aij]m×n and B = [bij]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) aij = bij ∀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=[aij]andB=[bij]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
 Cij=aij+bij ∀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 Om×n be an m × n Null Matrix and A=[aij] be also m× n matrix, then
A + Om×n =A= Om×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=[aij]m×n , there exist a matrix [– aij]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 = [aij]and B = [bij]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 = aij – bij ∀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 = [aij]m×n , we have
λA=[λaij]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 = [aij]m×n and B = [bij]n×p are two matrices of order m ×nand n ×prespectively, then their
product AB is of order m × pand is defined as

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