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Equality of Two Matrices:
If A = [aij] and B
= [bij]are the two matrices of same order then they
are said to be equal if their corresponding elements are same.
i.e., A = B if aij =bij
for all i and j.
Addition and Subtraction of Two Matrices:
= [aij]m ´ n and B =
[bij]m ´ n then
A + B = [aij + bij]m ´ n and A –
B= [aij – bij]m ´ n for all i
If A, B, C are three matrices of same order, then
A + B = B + A, Commutative
A + (B + C) = (A + B)
+ C, Associative law
A + 0 = 0 + A = A, where 0 is a
Multiplication of two Matrices:
If A = [aij]m ´ n and B =
[bjk]n ´ p are two
matrices then the product C = AB = [Cik]m ´ n where Cik
can be obtained by multiplying the ith row of A
and kth column of B.
Properties of Matrix Multiplication :
If A, B, C are three matrices, then
AB ¹ BA (in general)
A(BC) = (AB)C,
A(B + C) = AB
(A + B)C = AC + BC
A.0 = 0.A = 0, where 0 is a null matrix.
A.I = I.A = A, where I
is an identity matrix.
Properties of Transpose :
If A and B are two matrices and k is
a scalar, then
Some useful results:
Some useful properties: