INTRODUCTION TO MATRICES, TYPES OF MATRICES,
- 1. WrittenBy : AMIR HASSAN OFBS CHEMISTRY DEPARTMENT GPGC MARDAN INTRODUCATION TO MATRICES CONCEPTOF MATRIX: The concept of matrices was first prevented by Arther Kelley, an eminent mathematician, in the middle of 19th century, and its use in different scientific disciplines has since been increasing day by day. MATRICES IN CHEMISTRY: 1) Any symmetry operation about a symmetry element in a molecule involves the transformation of a set of coordinates x, y and z of an atom into a set of new coordinates x`, y` and z`. 2) The two sets of coordinates of the atom can be related by a set of equations. This set of equations may also be formulated in matrix notation. 3) Thus each symmetry operation can be represented by a specific matrix. 4) Knowledge of the matrices of various operations in a molecule will be useful to solve structural problems in chemistry. 5) For Example: The symmetry of vibrational modes in molecules can be analyzed using the matrices for different operation. MATRIX:- A matrix is a rectangular array of numbers or symbol for numbers arranged in rows and columns. i. In mathematical terms, Matrices (plural of matrix) are rectangular or, square tables whose elements are written in order in the form of rows and columns. ii. Matrices are usually represented by the capital letters of English alphabets i.e. A, B, C, D, E… and their numbers are shown by small letters of English alphabets i.e. a, b, c, d, e… iii. Horizontal entries in matrix are called its Rows, (denoted by i). iv. While vertical entries are called its columns, (denoted by j). v. Though it is not necessary that a matrix should have equal number of rows and columns, but it is necessary that number of members in different rows or columns should be equal. vi. For example: if the numbers in members in the first row of a matrix is 3 then the numbers of elements in all other rows will be 3. Similarly if there are 2 elements in first column of a matrix then all other columns of the same matrix will also have 2 elements. A= B= C= D= E= F= G= 1 -2 3 4 a b c d 5 7 x y 0 a aij bij cij dij
- 2. WrittenBy : AMIR HASSAN OFBS CHEMISTRY DEPARTMENT GPGC MARDAN ORDER OR, DIMENSIONOF A MATRIX:- Numbers of rows and columns in a matrix represent the order or dimension of a matrix. For Example: if the number of rows in a matrix is m and the number of columns be n then order or dimension of matrix will be m × n. Remember that order m × n does not mean the product of “m” and “n”. It’s read as “m by n”. i.e. order ofmatrix = no. ofrows × no. of columns A= and B= no. of rows is 2 × no. ofcolumn is also 2. Hence order ofmatrices A and B is 2 × 2. C= no. of rows is 1 × no. of column is 2. D= no. of rows is 2× no. of column is 1. Hence order of matrix C is 1 × 2. Hence order of matrix D is 2 × 1. KINDS OF A MATRICES:- 1) ROW MATRIX: A matrix consist of only one row and n column is called row matrix. Examples are the following: A= order is 1 × 4. B= order is 1 × 1. 2) COLUMN MATRIX: A matrix consists of only one column and m rows is called column matrix. Examples are the following : A= order is 2 × 1. B= order is 2 × 1. C= order is 1 × 1. 3) SQUARE MATRIX: A matrix in which the numbers of rows is equal to the number of column is called square matrix. Examples are the following: A= order is 2 × 2. B= order is 2 × 2. C= order is 1 × 1. 4) RECTANGULAR MATRIX: A matrix in which the numbers of rows is not equal to the number of column is called square matrix. A= no. of rows is 2 × no. of column is 1. D= no. of rows is 1× no. of column is 2. 1 -2 3 4 a b c d 5 7 x y 22 5 9 4 6 7 5 7 x y 2 1 -2 3 4 a b c d 2 5 7 x y
- 3. WrittenBy : AMIR HASSAN OFBS CHEMISTRY DEPARTMENT GPGC MARDAN 5) DIAGONAL MATRIX: A square matrix in which all the elements is zero “0”, except its diagonal elements is called diagonal matrix. Consider a square matrix elements a11,a22,a33,called its diagonal elements. A= M= N= R= 6) UNIT OR IDENTITYMATRIX: “A square matrix in which their diagonal element is equal to one (w.r.t multiplication) and every non diagonal element is equal to zero is called diagonal matrix”. It is usually denoted by I. Example are the following I= I= I= 7) ZERO/NULL MATRIX: A square or, rectangular matrix in which their all the element is equal to zero is called zero or null matrix. - It is usually denoted by letter “O” of English alphabet. Examples are the following O= O= O= O= 8) SCALAR MATRIX: A square matrix in which the diagonal elements are the same is called scalar matrix. - Examples are the following A= B= C= 9) NEGATIVE OF MATRIX: - “If the signs of all the elements of a matrix are changed then new matrix are formed is called negative of the matrix or, additive inverse of the matrix”. Examples are the following A= then -A= a11 a12 a13 a21 a22 a23 a31 a32 a33 1 0 0 1 x 0 0 y -1 0 0 4 1 0 0 1 1 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 2 -5 0 0 -5 1 0 0 1 2 -3 6 -1 -2 3 -6 1
- 4. WrittenBy : AMIR HASSAN OFBS CHEMISTRY DEPARTMENT GPGC MARDAN 10) TRANSPOSEOF MATRIX: The transpose of a matrix is obtained by interchanging its rows and columns are called transpose of the matrix. -It is usually denoted by At or Ã. If the order of matrix is m × n new matrix obtained order is n × m called transpose of the matrix. Example are the following IF A = then At or à = B= Bt = 11) ADJOINT OF MATRIX: The adjoint of a matrix is obtained by interchanging the principle diagonal elements positions and the signs of the second diagonal elements. A= Adj. A= A= Adj. A= 12) DETERMINANTOF MATRIX: The determinant of a matrix is obtained by multiplying the principal diagonal elements and subtracts the second diagonal elements. - The determinant of matrix contains exactly the same elements as its real matrix. The only difference is that of the elements of matrix are written inside a square bracket while in case of determinants two vertical line segments ׀׀ used instead of brackets. A= then, ׀A׀ = ׀A׀ = ad – bc 13) SINGULAR & NON SINGULAR MATRIX: If the value of determinant of square matrix is zero called singular matrix or not equal to zero known as non singular matrix. A= then ׀A=׀ ׀A =׀ 2 × 8 – 4 × 4 = 16 – 16 =0 (is singular matrix) B= then ׀B=׀ ׀ B =׀ 1 × 4 – 2 × 3 = 4 – 6 = – 2 ≠ 0 (is non singular matrix) -5 -7 2 1 -5 2 -7 1 9 7 9 7 a b c d d -b -c a 6 -3 4 -2 -2 3 -4 6 a b c d a b c d 2 4 4 8 2 4 4 8 1 2 3 4 1 2 3 4