This is a list of some of the properties of the set of real numbers that we need in order to work with vectors and matrices. Actually, we can work with matrices whose entries come from any set that satisfies these properties, such as the set of all rational numbers or the set of all complex numbers.
a + (b + c) = (a + b) + c and a . (b . c) = (a . b) . c.
a + b = b + a and a . b = b . a.
a . (b + c) = a . b + a . c and (a + b) . c = a . c + b . c.
a + 0 = a and 0 + a = a, and
a . 1 = a and 1 . a = a.
a + x = 0 and x + a = 0have a solution x in the set of real numbers, called the additive inverse of a, denoted by -a.
a . x = 1 and x . a = 1have a solution x in the set of real numbers, called the multiplicative inverse of a, denoted by a-1.
Here are some additional properties of real numbers a,b,c, which can be proved from the properties listed above.