# 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).