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Scalar Multiplication of a Matrix

The operation of multiplying variables by a constant scalar factor may properly be called scalar multiplication and the rule of multiplication of matrix by a scalar is that
the product of an m × n matrix A = [aij] by a scalar quantity c is the m × n matrix [bij] where bij = caij.

It is denoted by cA or Ac

For example:

c [a11a12a13a21a22a23a31a32a33]

= [ca11ca12ca13ca21ca22ca23ca31ca32ca33]

= [a11ca12ca13ca21ca22ca23ca31ca32ca33c]

= [a11a12a13a21a22a23a31a32a33] c.

The product of an m × n matrix A = (aij)m, n by a scalar k where k ∈ F, the field of scalars, is a matrix B = (bij)m, n defined by bij = kaij, i = 1, 2, 3, ....., m : j = 1, 2, 3, ....., n and is written as B = kA.

Let A be an m × n matrix and k, p are scalars. Then the following results are obvious.

(i) k(pA) = (kp)A,

(ii) 0A = Om, n,

(iii) kOm, n = Om, n,

(iv) kIn = [k0...00k...0............00...k],

(v) 1A = A, where 1 is the identity element of F.

The scalar matrix of order n whose diagonal elements are all k can be expressed as kIn.

In general, if c is any number (scalar or any complex number) and a is a matrix of order m × n, then the matrix cA is obtained by multiplying each element of the matrix A by the scalar c.

In other words, A = [aij]m × n

then, cA = [kij]m × n, where kij = caij


Examples on scalar multiplication of a matrix:

1. If A = [3120] and c = 3, then

cA = 3[3120]

    = [3×33×13×23×0]

    = [9360]


2. If A = [015321204] and c = -5, then

cA = -5[015321204]

     = [(5)×0(5)×(1)(5)×5(5)×(3)(5)×2(5)×1(5)×2(5)×0(5)×(4)]

     = [05251510510020]





10th Grade Math

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