Matrix Arithmetic

Part 6: Summary

In the list below we tabulate some of the properties of arithmetic with real numbers. For each property indicate in your worksheet whether the corresponding property holds for matrices.

1. Associativity of addition:
   
   
   • (a + b) + c = a + (b + c), for all a, b, c

2. Distributivity of multiplication over addition:
   
   
   • a(b + c) = ab + ac, for all a, b, c

3. Commutativity of multiplication:
   
   
   • ab = ba, for all a and b

4. Associativity of multiplication:
   
   
   • (ab)c = a(bc), for all a, b, c

5. Distributivity of exponentiation over multiplication:
   
   
   • (ab)^k = a^kb^k, for all real numbers a and b and positive integers k

6. Multiplicative identity:
   
   
   • 1 x a = a, for all a

7. Multiplicative zero:
   
   
   • 0 x a = 0, for all a

8. Absence of "zero divisors":
   
   
   • If ab = 0, then a = 0 or b = 0.

9. Multiplicative cancellation:
   
   
   • If ab = ac and a is not zero, then b = c.

We have seen some properties that are peculiar to matrices. Complete the statements below with either correct symbols or with a verbal description. Assume that A is an m x n matrix, B is an n x p matrix, and x is an n-dimensional vector.

10. (AB)^T =
11. Ae_j =
12. Ax can be written as a linear combination of vectors as follows:

Two matrix multiplicative properties that hold for diagonal matrices, but not for matrices in general, are the following:

13. Multiplication of diagonal matrices is... (fill in a property)
14. A power of a diagonal matrix can be computed by ...