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Operations on Matrices - Properties of Scalar Multiplication of a Matrix

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notes

If A = `[a_(ij)]` and B = `[b_(ij)]` be two matrices of the same order, say m × n, and k and l are scalars, then 
(i) k(A +B) = k A + kB, 

(ii) (k + l)A = k A + l A

(iii) k (A + B) = k `([a_(ij)]` + `[b_(ij)]`)

= k `[a_(ij) + b_(ij)]` = `[k (a_(ij) + b_(ij))]` = `[(k a_(ij)) + (k b_(ij))]` 
= `[k a_(ij)] + [k b_(ij)] = k [a_(ij)] + k [b_(ij)]` = kA + kB 

(iv) (k + l) A = (k + l) `[a_(ij)]` 
= [(k + l) `a_(ij)`] + [k `a_(ij)`] + [l `a_(ij)`] 
= k [`a_(ij)`] + l [`a_(ij)`] 
= k A + l A

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