Matrices and Determinants, Class 12, Mathematics (IIT) Chapter Notes PDF Download

Definition of Matrix and Types of Matrices - Matrices and Determinants, Class 12, Mathematics

A. DEFINITION
Any rectangular arrangement of numbers (real or complex) (or of real valued or complex valued expressions) is called a matrix. If a matrix has m rows and n columns then the order of matrix is said to be m by n (denoted as m × n).
The general m × n matrix is A =

where aij denote the element of i^th row & j^th column. The above matrix is usually denoted as [a_ij]_{m × n}

Note :
(i) The elements a₁₁, a₂₂, a₃₃,....... are called as diagonal elements. Their sum is called as trace of A denoted as tr(A)
(ii) Capital letters of English alphabets are used to denote matrices.

B. TYPES OF MATRICES

(i) Row Matrix : A matrix having only one row is called as row matrix (or row vector).
General form of row matrix is A =

(ii) Column Matrix : A matrix having only one column is called as column matrix
(or column vector). General form of A =

(iii) Square Matrix : A matrix in which number of rows & columns are equal is called a square
matrix. The general form of a square matrix is A =

(iv) Zero Matrix : A =

(v) Upper Triangular Matrix :

A =
(i.e., all the elements below the diagonal element are zero.)

(vi) Lower Triangular Matrix :

A =
(i.e., all the elements above the diagonal elements are zero.)

(vii) Diagonal matrix :
(i.e., all the elements of the square matrix other than diagonal elements are zero)
Note : Diagonal matrix of order in is denoted as Diag (a₁₁, a₂₂,....... a_nn).

(viii) Scalar Matrix : Scalar matrix is a diagonal matrix in which all the diagonal elements are same.

(ix) Unit Matrix (Identity Matrix) : Unit matrix is a diagonal matrix in which all the diagonal elements
are unity. Unit matrix of order ‘n’ is denoted y In (or I).