APP. A] VECTORS AND MATRICES 415
Both are square matrices. The first is of order 3, and its diagonal consists of the elements 1,− 4 ,2 so its trace
equals 1− 4 + 2 =−1. The second matrix is of order 4; its diagonal consists only of 1’s, and there are only 0’s
elsewhere. Thus the second matrix is the unit matrix of order 4.
Algebra of Square Matrices
LetAbe any square matrix. Then we can multiplyAby itself. In fact, we can form all nonnegativepowers
ofAas follows:
A^2 =AA, A^3 =A^2 A, ..., An+^1 =AnA, ..., and A^0 =I(whenA= 0 )Polynomials in the matrixAare also defined. Specifically, for any polynomial
f(x)=a 0 +a 1 x+a 2 x^2 +···+anxnwhere theaiare scalars, we definef (A)to be the matrix
f (A)=a 0 I+a 1 A+a 2 A^2 +···+anAnNote thatf (A)is obtained fromf(x)by substituting the matrixAfor the variablexand substituting the scalar
matrixa 0 Ifor the scalar terma 0. In the case thatf (A)is the zero matrix, the matrixAis then called azeroor
rootof the polynomialf(x).
EXAMPLE A.7 SupposeA=
[
12
3 − 4]. Then
A^2 =[
12
3 − 4][
12
3 − 4]
=[
7 − 6
− 922]
andA^3 =A^2 A=[
7 − 6
− 922][
12
3 − 4]
=[
− 11 38
57 − 106]Supposef(x)= 2 x^2 − 3 x+5. Then
f (A)= 2[
7 − 6
− 9 − 22]
− 3[
12
3 − 4]
+ 5[
10
01]
=[
16 − 18
−27 61]Supposeg(x)=x^2 + 3 x−10. Then
g(A)=[
7 − 6
− 922]
+ 3[
12
3 − 4]
− 10[
10
01]
=[
00
00]ThusAis a zero of the polynomialg(x).
A.8Invertible (Nonsingular) Matrices, Inverses
A square matrixAis said to beinvertible(ornonsingular) if there exists a matrixBsuch thatAB=BA=I, (the identity matrix).Such a matrixBis unique; it is called theinverseofAand is denoted byA−^1. Observe thatBis the inverse ofA
if and only ifAis the inverse ofB. For example, suppose
A=[
25
13]
and B=[
3 − 5
− 12]