Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

414 VECTORS AND MATRICES [APP. A

Matrix multiplication does, however, satisfy the following properties:

Theorem A.2: LetA,B,Cbe matrices. Then, whenever the products and sums are defined:

(i) (AB)C=A(BC)(Associative Law).

(ii) A(B+C)=AB+AC (Left Distributive Law).

(iii)(B+C)A=BA+CA(Right Distributive Law).

(iv) k(AB)=(kA)B=A(kB)wherekis a scalar.

Matrix Multiplication and Systems of Linear Equations

Any systemSof linear equations is equivalent to the matrix equation

AX=B

whereAis the matrix consisting of the coefficients,Xis the column vector of unknowns, andBis the column
vector of constants. (Hereequivalentmeans that any solution of the systemSis a solution to the matrix equation
AX=B, and vice versa.) For example, the system

x+ 2 y− 3 z= 4

5 x− 6 y+ 8 z= 9

is equivalent to

[

12 − 3

5 − 68

]

⎡

⎣

x

y

z

⎤

⎦=

[

4

9

]

Observethatthe system is completely determined by the matrix

M=[A, B]=

[

12 − 34

5 − 689

]

which is called theaugmented matrixof the system.

A.6Transpose

Thetransposeof a matrixA, writtenAT, is the matrix obtained by writing the rows ofA, in order, as columns.
For example,
[
123
456

]T

=

⎡

⎣

14

25

36

⎤

⎦ and [ 1 ,− 3 ,− 5 ]T=

⎡

⎣

1

− 3

− 5

⎤

⎦

Note that ifAis anm×nmatrix, thenATis ann×mmatrix. In particular, the transpose of a row vector is a column
vector, and vice versa. Furthermore, ifB=[bij]is the transpose ofA=[aij], thenbij=ajifor alliandj.

A.7Square Matrices

A matrix with the same number of rows as columns is called asquare matrix. A square matrix withnrows
andncolumns is said to be ofordern, and is called ann-square matrix.
The main diagonal, or simply diagonal,ofann-square matrixA=[aij]consists of the elements
a 11 ,a 22 ,...,ann, that is, the elements from the upper left corner to the lower right corner of the matrix. The
traceofA, written tr(A), is the sum of the diagonal elements, that is, tr(A)=a 11 +a 22 +···+ann.
Then-squareunit matrix, denoted byIn, or simplyI, is the square matrix with 1’s along the diagonal and
0’s elsewhere. The unit matrixIplays the same role in matrix multiplication as the number 1 does in the usual
multiplication of numbers. Specifically, for any matrixA,

AI=IA=A

Consider, for example, the matrices

⎡

⎣

1 − 20

0 − 4 − 6

532

⎤

⎦ and

⎡

⎢

⎢

⎣

1000

0100

0010

0001

⎤

⎥

⎥

⎦