4 Multiplication of a matrix by a number

There is also a natural way of defining the product of a matrix with a number. Using the matrix A above, we note that

A + A = 1 2 3 4 + 1 2 3 4 = 2 4 6 8

What we see is that 2 A (which is the shorthand notation for A + A ) is obtained by multiplying every element of A by 2 .

In general if A is an m × n matrix with typical element a i j then the product of a number k with A is written k A and has the corresponding elements k a i j .

Hence, again using the matrix A above,

7 A = 7 1 2 3 4 = 7 14 21 28

Similarly:

3 A = 3 6 9 12

Task!

For the following matrices find, where possible, A + B , A B , B A , 2 A .

  1. A = 1 2 3 4 B = 1 1 1 1
  2. A = 1 2 3 4 5 6 7 8 9 B = 1 1 1 1 1 1 1 1 1
  3. A = 1 2 3 4 5 6 7 8 9 B = 1 2 3 4 5 6
  1. A + B = 2 3 4 5 A B = 0 1 2 3 B A = 0 1 2 3 2 A = 2 4 6 8
  2. A + B = 2 3 4 3 4 5 8 9 10 A B = 0 1 2 5 6 7 6 7 8 B A = 0 1 2 5 6 7 6 7 8

    2 A = 2 4 6 8 10 12 14 16 18

  3. None of A + B , A B , B A , are defined. 2 A = 2 4 6 8 10 12 14 16 18