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Matrix scalar multiplication

Definition.
The product of the matrix A to number k is a matrix B = k · A of the same size derived from matrix A by multiplying every entry of A by k:

bi,j = k · ai,j


Properties of matrix scalar multiplication

  • 1 · A = A
  • 0 · A = Θ, where Θ - zero matrix
  • k · (A + B) = k · A + k · B
  • (k + n) · A = k · A + n · A
  • (k · n) · A = k · (n · A)

Examples of matrix scalar multiplication

Example 1.
Find the product of matrix A =  (  4  2  )  and the number 5.
 9  0 

Solution:

5·A=  (  4  2  )  =  (  5·4  5·2  )  =  (  20  10  )
 9  0   5·9  5·0   45  0 
Example 2
Find the product of matrix A =  (  2  -2  )  and the number (-2).
 -1  0 
 5  -1 

Solution:

(-2)·A = (-2)· (  2  -2  )  =  (  (-2)·2  (-2)·(-2)  )  =  (  -4  4  )
 -1  0   (-2)·(-1)  (-2)·0   2  0 
 5  -1   (-2)·5  (-2)·(-1)   -10  2 

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