α
β
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−α
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Rationalised 2023-24
 74
MATHEMATICS
® kA = k[aij]m × n = [k(aij)]m × n
® – A = (–1)A
® A – B = A + (–1) B
® A + B = B + A
® (A + B) + C = A + (B + C), where A, B and C are of same order ® k(A + B) = kA + kB, where A and B are of same order, k is constant ® (k + l ) A = kA + lA, where k and l are constant ®® If A = [aij]m × n and B = [bjk]n × p, then AB = C = [cik]m × p, where      =
(i) A(BC) = (AB)C,  (ii)   A(B + C) = AB + AC,  (iii)   (A + B)C = AC + BC
®®  If A = [aij]m × n, then A′ or AT = [aji]n × m
(i) (A′)′ = A,  (ii)   (kA)′ = kA′,  (iii)   (A + B)′ = A′ + B′,  (iv)  (AB)′ = B′A′
® A is a symmetric matrix if A′ = A