Maths > Matrices and Determinants > 1.0 Introduction
Matrices and Determinants
1.0 Introduction
2.0 Algebra of Matrices
3.0 Special Matrices
3.1 (a) Symmetric Matrix:
3.2 (b) Skew Symmetric Matrix:
3.3 (c) Hermitian matrix:
3.4 (d) Skew Hermitian Matrix:
3.5 (e) Singular and Non-singular Matrices:
3.6 (f) Orthogonal Matrix :
3.7 (g) Idempotent Matrix :
3.8 (h) Unitary Matrix :
3.9 (i) Involuntary Matrix:
3.10 (j) Nilpotent Matrix:
4.0 Determinant of a square matrix
5.0 Adjoint of a square Matrix
6.0 Inverse of a Matrix
7.0 Types of Equations Homogenous & Non-Homogenous
8.0 Cramer's rule
9.0 Types of Linear Equations
1.1 Type of Matrices
3.2 (b) Skew Symmetric Matrix:
3.3 (c) Hermitian matrix:
3.4 (d) Skew Hermitian Matrix:
3.5 (e) Singular and Non-singular Matrices:
3.6 (f) Orthogonal Matrix :
3.7 (g) Idempotent Matrix :
3.8 (h) Unitary Matrix :
3.9 (i) Involuntary Matrix:
3.10 (j) Nilpotent Matrix:
(a) Row Matrix :
A Matrix having a single row is called a row matrix.
Example:
$A = \left[ {\begin{array}{c} 4&1&7 \end{array}} \right]$ is a row matrix of order 1 x 3
(b) Column Matrix:
A Matrix having a single column is called a column matrix.
Example:
$A = \left[ {\begin{array}{c} 9 \\ 1 \\ 7 \end{array}} \right]$ is a column matrix of order 3 x 1
(c) Square Matrix:
A matrix in which the number of rows is equal to the number of columns is called a square matrix.
Example:
$A = \left[ {\begin{array}{c} 1&4 \\ 7&5 \end{array}} \right]$ is a square matrix of order 2 x 2
(d) Diagonal Matrix:
A square matrix whose non diagonal elements are zero is called a diagonal matrix.
Example:
$A = \left[ {\begin{array}{c} a&0&0 \\ 0&b&0 \\ 0&0&c \end{array}} \right]$ is a diagonal matrix of order 3 x 3
(e) Scalar Matrix:
A scalar matrix is a diagonal matrix whose diagonal elements are equal.
Example:
$A = \left[ {\begin{array}{c} k&0&0 \\ 0&k&0 \\ 0&0&k \end{array}} \right]$ is a scalar matrix of order 3 x 3
(f) Unit or Identity Matrix:
A unit matrix is a diagonal matrix whose diagonal elements are all equal to one.It is denoted by ${I_n}$ where $n$ is the order of the square matrix.
Example:
$A = \left[ {\begin{array}{c} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right]$ is a unit matrix of order 3 x 3
(g) Null or Zero Matrix:
If all the elements of a matrix are zero,then it is called null matrix.
Example:
$A = \left[ {\begin{array}{c} 0&0&0 \\ 0&0&0 \end{array}} \right]$ is a zero matrix of order 2 x 3
(h) Upper Triangular Matrix:
A square matrix in which all the elements below the principal diagonal are zero is called upper triangular matrix.
Example:
$A = \left[ {\begin{array}{c} 3&{ - 1}&8 \\ 0&2&{ - 3} \\ 0&0&6 \end{array}} \right]$ is an upper triangular matrix of order 3 x 3
(i) Lower Triangular Matrix:
A square matrix in which all the elements above the principal diagonal are zero is called lower triangular matrix.
Example:
$A = \left[ {\begin{array}{c} 6&0&0&0 \\ { - 1}&4&0&0 \\ 2&9&7&0 \\ { - 4}&1&2&6 \end{array}} \right]$ is a lower triangular matrix of order 4 x 4
Note:-
(i) A diagonal matrix is both upper and lower triangular matrix.
(ii)Diagonal,Unit,Scalar,Triangular matrices are square matrices.
(iii)Principal diagonal is the largest diagonal in the square matrix.
