1st Year Math Notes Chapter 3
Handwritten Important Short Notes of 1st Year Math Notes Chapter 3 written by Professor Asad Khalid Suib. These notes are very helpful in the preparation of Matrices and Determinants for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.
Summary and Contents:
Topics which are discussed in the notes are given below:
- Our comprehensive 1st Year Math Notes Chapter 3 will ensure you're fully prepared for your exams.
- Definition of unit Matrix with examples
- Definition of column Matrix with example
- Definition of a symmetric matrix with examples
- Definition of skew-symmetric matrix with examples
- Definition of a diagonal Matrix with examples
- Transpose of a matrix with example
- Cofactor of an element with an example
- Square Matrix with example
- While solving linear systems of equations, a new notation was introduced to reduce the amount of writing. For this new notation the word matrix was irst used by the English mathematician James Sylvester (1814 - 1897). Arthur Cayley (1821 - 1895) developed the theory of matrices and used them in the linear transformations. Now-a-days, matrices are used in high speed computers and also in other various disciplines. The concept of determinants was used by Chinese and Japanese but the Japanese mathematician Seki Kowa (1642 - 1708) and the German Mathematician Gottfried Wilhelm Leibniz (1646 - 1716) are credited for the invention of determinants. G. Cramer (1704 - 1752) applied the determinants successfully for solving the systems of linear equations. A rectangular array o f numbers enclosed by a pair o f brackets such as: [ 3 4 5 ] is called a matrix. The horizontal lines of numbers are called rows and the vertical lines of numbers are called columns. The numbers used in rows or columns are said to be the entries or elements of the matrix. The matrix in (i) has two rows and three columns while the matrix in (ii) has 4 rows and three columns. Note that the number of elements of the matrix in (ii) is 4 % 3 = 12. Now we give a general deinition of a matrix. Generally, a bracketed rectangular array of m%n elements. aij (i = 1, 2, 3, ...., m; j = 1, 2, 3, ...., n), arranged in m rows and n columns is called an m by n matrix (written as m % n matrix). m % n is called the order of the matrix in (iii). We usually use capital letters such as A, B, C, X, Y, etc., to represent the matrices and small letters such as a, b, c,... I, m, n,...,a11, a12, a13,...., etc., to indicate the entries of the matrices. Let the matrix in (iii) be denoted by A. The ith row and the jth column of A are indicated in the following tabular representation of A.
- The elements of the ith row o f A are ai1 ai2 ai3...... aij...... ain while the elements of the jth column of A are a1j a2j a3j.... aij...... amj. We note that aij is the element of the ith row and jth column of A. The double subscripts are useful to name the elements of the matrices. For example, the element 7 is at a23 position in the matrix. A = [aij]m%n or A = [aij], for i = 1, 2, 3,...., m; j = 1, 2, 3,...., n, where aij is the element of the ith row and jth column of A.
- Note: aij is also known as the (i, j)th element or entry of A.
- Note: The matrix A is called real if all of its elements are real.
- Row Matrix or Row vector: A matrix, which has only one row, i.e., a 1 % n matrix of the form [ai1 ai2 ai3 ...... ain] is said to be a row matrix or a row vector.
- Column Matrix or Column Vector: A matrix which has only one column i.e., an m % 1 matrix of the form is said to be a column matrix or a column vector.