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 = [a_ij] and B = [b_ij] are two m × n matrices.

I. k(A + B) = k([a_ij] + [b_ij])

= k[a_ij + b_ij], (by using the definition of addition of matrices)

= [k(a_ij + b_ij)], (by using the definition of scalar multiplication of matrices)

= [ka_ij + kb_ij]

= [ka_ij] + [kb_ij]

= k[a_ij] + k[b_ij]

= kA + kB

Therefore, k(A + B) = kA + kB (proved).

II. (k + c)A = (k + c) [a_ij]

= [(k + c) (a_ij)], (by using the definition of scalar multiplication of matrices)

= [ka_ij + ca_ij]

= [ka_ij] + [ca_ij]

= k[a_ij] + c[a_ij]

= kA + cA

Therefore, (k + c)A = kA + cA (proved).

III. k(cA) = k(c[a_ij])

= k[ca_ij], (by using the definition of scalar multiplication of matrices)

= [k(ca_ij)]

= [(kc) a_ij], (by using the definition of scalar multiplication of matrices)

= (kc) [a_ij]

= (kc)A

Therefore, k(cA) = (kc)A (proved).

IV. 1A = 1[a_ij]

= [1 ∙ a_ij]

= [a_ij]

= A

Therefore, 1A = A (proved).

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