The trace of a matrix
The trace of a n×nsquare matrix A is denoted Tr(A)and represents a summation
of the diagonal elements of the matrix A. One can write
Tr(A) =
∑ni=1aii =a 11 +a 22 +a 33 +···+annIf matrices Aand Bare conformable matrices, then the trace satisfies the properties
Tr(A+B) = Tr(A) + Tr (B), Tr(AB ) = Tr(BA )
The Inverse Matrix
If A and E are square matrices such that their matrix product produces the
identity matrix, that is, if AE =EA =I, then E is called the inverse of Aand the
matrix Eis written
E=A−^1 ,which is read “Eequals Ainverse”. Thus, the inverse matrix has the property that
AA −^1 =A−^1 A=I.The inverse matrix, if it exists, is unique. This statement can be proven by first
assuming that the inverse is not unique and then showing that this assumption is
wrong. This type of proof is known as the method of reductio ad absurdum^1 to
verify something is true.
For example, if A 1 and A 2 are both inverses of the matrix A, then by hypothesis
both of the statements
AA 1 =A 1 A=I and AA 2 =A 2 A=I
must be true. Consequently, one can write
A 2 =A 2 I=A 2 (AA 1 ) = (A 2 A)A 1 =IA 1 =A 1.Hence, A 2 =A 1 =A−^1 and the initial assumption is wrong and so the inverse matrix
must be unique.
(^1) The method of reductio ad absurdum is used to prove a statement in mathematics by assuming initially that
the statement is true (or false) and then performing an analysis of this assumption (the reduction of the proposition)
to arrive at a conclusion which is obviously absurd and contradicts the initial assumption. The method of reductio
ad absurdum was used by the early Greek mathematicians as a method for proving many theorems.