Advanced Inorganic Chemistry/Matrix (1.3)

A matrix is a rectangular array of quantities or expressions in rows (m) and columns (n) that is treated as a single entity and manipulated according to particular rules. The dimension of a matrix is denoted by m × n. In inorganic chemistry, molecular symmetry can be modeled by mathematics by using group theory. The internal coordinate system of a molecule may be used to generate a set of matrices, known as a representation, that corresponds to particular symmetry operations. Matrix modeling thus allows for symmetry operations performed on the molecule to be represented in an identical fashion mathematically.

Addition
The sum of two matrices, A and B, is carried out by adding or subtracting the element of one matrix with the corresponding element of another matrix. These operations may only be performed on matrices of identical dimension.

A + B = $$\sum_{i=1}^{m}(\sum_{j=1}^n A_{ij} + B_{ij})$$  where i refers to a particular row and j to a particular column.

Example:

$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix} + $$ $$\begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \end{pmatrix} = $$ $$\begin{pmatrix} {a_{11}+b_{11}} & {a_{12}+b_{12}} & {a_{13}+b_{13}} \\ {a_{21}+b_{21}} & {a_{22}+b_{22}} & {a_{23}+b_{23}} \end{pmatrix}$$

Scalar Multiplication
Multiplication of a matrix by a scalar, c, multiplies every element within the matrix by the scalar.

cA = c · Ai,j

Example:

c $$\cdot$$ $$ \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} = \begin{pmatrix} c\cdot a_{11} & c\cdot a_{12} \\ c\cdot a_{21} & c\cdot a_{22} \end{pmatrix}$$

Matrix Multiplication
Matrix multiplication entails computing the dot product of the row of one matrix, A, with the column of another matrix, B. Matrix multiplication is only defined if the number if columns of A, denoted by n, is equal to the number of rows of B, denoted by m. Their product is then the m × n matrix, C. Matrix multiplication entails some mathematical properties. First, it is associative; in other words, (A × B) × C = A × (B × C). Furthermore, matrix multiplication is not commutative; in other words, A × B =/= B × A

Cm×n = Am×c · Bc×n = $$\sum_{k=1}^c A_{i,k}B_{k,j}$$

Example:

$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{pmatrix} \times\begin{pmatrix} b_{11} \\ b_{21} \\ b_{31} \end{pmatrix} = \begin{pmatrix} {a_{11}b_{11} + a_{12}b_{21} + a_{13}b_{31}}  \\ {a_{21}b_{11} + a_{22}b_{21} + a_{23}b_{31}} \end{pmatrix}$$

Row Operations
There are three kinds of elementary row operations that are used to transform a matrix:

Square Matrices
Square matrices are matrices where the number of rows and number of columns are equal, resulting in an n × n matrix.

Identity Matrix
The identity matrix, In, is a diagonal matrix which has all elements along the main diagonal equal to 1 and all other elements equal to 0. Multiplication of another matrix by the identity matrix leaves the first unchanged. Moreover, multiplication with the identity matrix is commutative; in other words, A × I = I × A.

Example:

A·I3 = $$\begin{pmatrix} a & b & c \\ d & e & f \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} a & b & c \\ d & e & f\end{pmatrix} $$

Trace
Only applicable to square matrices, the trace or character, $$\chi $$, of a matrix is the sum of its diagonal entries along the main diagonal.

Determinant
The determinant of a matrix, denoted det(A), is a real number computed from a square matrix. A non-zero determinant implies matrix invertibility, which further implies that the set of linear equations comprising the matrix has exactly one solution.

For a 2 × 2 matrix, the determinant is computed as follows:

det(A) = $$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc $$

For a 3 × 3 matrix, the determinant is computed as follows:

det(A) = $$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a\begin{vmatrix} e & f \\ h & i \end{vmatrix} -b\begin{vmatrix} d & f \\ g & i \end{vmatrix}+c\begin{vmatrix} d & e \\ g & h \end{vmatrix} $$

Higher order determinants may be calculated by using Cramer's Rule.