Properties of Scalar Multiplication of a Matrix
We will discuss about the properties of scalar multiplication of a matrix.
If X and Y are two m × n matrices (matrices of the same order) and k, c and 1 are the numbers (scalars). Then the following results are obvious.
I. k(A + B) =
kA + kB
II. (k + c)A = kA + cA
III. k(cA) = (kc)A
IV. 1A = A
Proof: Let A = [aij] and B = [bij] are two m × n matrices.
I. k(A + B) = k([aij] + [bij])
= k[aij + bij], (by using the definition of addition of matrices)
= [k(aij + bij)], (by using the definition of scalar multiplication of matrices)
= [kaij + kbij]
= [kaij] + [kbij]
= k[aij] + k[bij]
= kA + kB
Therefore, k(A + B) = kA + kB (proved).
II. (k + c)A = (k + c) [aij]
= [(k + c) (aij)], (by using the definition of scalar multiplication of matrices)
= [kaij + caij]
= [kaij] + [caij]
= k[aij] + c[aij]
= kA + cA
Therefore, (k + c)A = kA + cA (proved).
III. k(cA) = k(c[aij])
= k[caij], (by using the definition of scalar multiplication of matrices)
= [k(caij)]
= [(kc) aij], (by using the definition of scalar multiplication of matrices)
= (kc) [aij]
= (kc)A
Therefore, k(cA) = (kc)A (proved).
IV. 1A = 1[aij]
= [1 ∙ aij]
= [aij]
= A
Therefore, 1A = A (proved).
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