Physical Chemistry Third Edition

B Some Useful Mathematics 1249

B.7 Matrices and Determinants

Amatrixis an array or list of quantities arranged in rows and columns. If the matrixA

hasmrows andncolumns, it is called anmbynmatrix and is written in the form:

A

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

A 11 A 12 A 13 ··· A 1 n

A 21 A 22 A 23 ··· A 2 n

A 31 A 32 A 33 ··· A 3 n

··· ··· ··· ··· ···

Am 1 Am 2 Am 3 ··· Amn

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(B-80)

where the ellipses (...) indicate additional entries. The entries in the two-dimensional

list are calledelementsof the matrix. Ifmn, the matrix is asquare matrix.Two

matrices are equal to each other if both have the same number of rows and the same

number of columns and if every element of one is equal to the corresponding element

of the other. Three-dimensional (and higher) matrices also exist, but we will not need

to use them.

Matrix Algebra If the matrixCis the sum ofAandB, it is defined by

CijAij+Bij for everyiandj (B-81)

The matricesA,B, andCmust have the same number of rows and the same number

of columns for the addition to be valid.

The product of a matrixAand a scalarcis denoted byBcAand defined by

BijcAij for everyiandj (B-82)

The product of two matrices is similar to the scalar product of two vectors as written in

Eq. (B-35). Let the components of two vectors be calledF 1 ,F 2 ,F 3 , andG 1 ,G 2 ,G 3.

instead ofFx,Fy, etc. Eq. (B-35) is the same as

F·GF 1 G 1 +F 2 G 2 +F 3 G 3 

∑^3

k 1

FkGk (B-83)

We define matrix multiplication in a similar way. IfA,B, andCare matrices such that

Cis the productAB, then

Cij

∑n

k 1

AikBkj (B-84)

wherenis the number of columns inA, which must equal the number of rows in the

matrixB. The matrixCwill have as many rows asAand as many columns asB.

We can think of the vectorFin Eq. (B-83) as a matrix with one row and three

columns (arow vector) and the vectorGas being a matrix with three rows and one

column (acolumn vector). Equation (B-83) is then a special case of Eq. (B-84):

F·G[F 1 F 2 F 3 ]

⎡

⎣

G 1

G 2

G 3

⎤

⎦ (B-85)