5 Some simple matrix properties

Using the definition of matrix addition described above we can easily verify the following properties of matrix addition:

Key Point 2

Basic Properties of Matrices

Matrix addition is commutative : A + B = B + A

Matrix addition is associative : A + ( B + C ) = ( A + B ) + C

The distributive law holds: k ( A + B ) = k A + k B

These Key Point results follow from the fact that a i j + b i j = b i j + a i j etc.

We can also show that the transpose of a matrix satisfies the following simple properties:

Key Point 3

Properties of Transposed Matrices

( A + B ) T = A T + B T ( A B ) T = A T B T ( A T ) T = A
Example 3

Show that ( A T ) T = A for the matrix   A = 1 2 3 4 5 6

Solution

A T = 1 4 2 5 3 6 so that ( A T ) T = 1 2 3 4 5 6 = A

Task!

For the matrices A = 1 2 3 4 , B = 1 1 1 1 verify that

(i)  3 ( A + B ) = 3 A + 3 B (ii)  ( A B ) T = A T B T .

(i)  A + B = 2 1 2 5 ; 3 ( A + B ) = 6 3 6 15 ; 3 A = 3 6 9 12 ;

3 B = 3 3 3 3 ; 3 A + 3 B = 6 3 6 15 .

(ii)  A B = 0 3 4 3 ; ( A B ) T = 0 4 3 3 ; A T = 1 3 2 4 ;

B T = 1 1 1 1 ; A T B T = 0 4 3 3 .

Exercises
  1. Find the coefficient matrix A of the system: 2 x 1 + 3 x 2 x 3 = 1 4 x 1 + 4 x 2 = 0 2 x 1 x 2 x 3 = 0

    If B = 1 2 3 4 5 6 0 0 1 determine ( 3 A T B ) T .

  2. If A = 1 2 3 4 5 6 and B = 1 4 0 1 2 7 verify that 3 ( A T B ) = ( 3 A 3 B T ) T .
  1. A = 2 3 1 4 4 0 2 1 1 , A T = 2 4 2 3 4 1 1 0 1 , 3 A T = 6 12 6 9 12 3 3 0 3

    3 A T B = 5 10 3 5 7 9 3 0 4 ( 3 A T B ) T = 5 5 3 10 7 0 3 9 4
  2. A T = 1 4 2 5 3 6 , A T B = 2 0 2 4 1 1 , 3 ( A T B ) = 6 0 6 12 3 3

    B T = 1 0 2 4 1 7 , 3 A 3 B T = 3 6 9 12 15 18 3 0 6 12 3 21 = 6 6 3 0 12 3