Linear Algebra
Chapter 3: Reduced Row Echelon Form
Chapter 4: Rank of a Matrix
Chapter 5: Solution of System of Equations
Chapter 6: Eigenvalues and Eigenvactors
Other chapters will be updated soon.
Chapter 1: Matrix Algebra
In this chapter you will study about
Matrix, Types of Matrices, symmetric, skew symmetric, Hermitian, skew hermitian etc.
1.2. Matrix: A matrix is a rectangular arrangement of numbers, symbols, or expressions, organized into rows and columns.
Matrices are used extensively in various fields, including computer graphics, physics, economics, and engineering, to solve systems of equations and represent data.
Examples:
Here A, B, C, D are the examples of matrix. The
vertical arrangement of data is called Column and horizontal presentation of
data is called Row. The order of matrix is represented by 
the number of
rows and
the number of
columns.
Thus, the order
of matrix A is 
The order of
matrix B is 
the order of
matrix C is 
and the order
of matrix D is 
.
An m × n matrix: the m rows are horizontal and the n columns are vertical. Each element of a matrix is often denoted by a variable with two subscripts. For example, a2,1 represents the element at the second row and first column of the matrix.
1.2. Type of Matrices:
i. Row Matrix: A matrix with one row, sometimes used to represent a vector.
Example: 
is a row matrix of order 
.
ii. Column Matrix: A matrix with one column is called column matrix.
Example: 
is a column
matrix of order 
.
iii. Square Matrix: A matrix with the same number of rows and columns, sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation, or shearing.
Example: 
iv. Rectangular Matrix: A matrix with different number of rows and columns is called rectangular Matrix.
v. Diagonal Matrix: A square matrix, whose all non-diagonal elements are zero and at least one diagonal element is non-zero, is called a diagonal Matrix.
Example: 
vi. Unit or Identity Matrix: a diagonal matrix, whose diagonal elements are unity, is called a unit or identity matrix and is denoted by I.
Example: 
vii.
Zero or Null Matrix: A matrix whose each element is zero is called a Zero
Matrix or Null Matrix. A zero matrix is usually denoted by 
.
Example:
viii. Scalar Matrix: A diagonal matrix, whose all-diagonal elements are equal, is called a scalar matrix.
Example: 
ix. Upper Triangular Matrix: A square matrix, whose all the elements below the principal diagonal are zero, is called an upper triangular matrix.
Example: 
x. Lower Triangular Matrix: A square matrix, whose all the elements above the principal diagonal are zero, is called an Lower Triangular Matrix,
Example: 
1.3. Equality of Matrices: Two 
matrices A and
B are said to be equal if the corresponding elements of the two matrices are
equal.
Thus, the two matrices 
and 
are equal if
i.
A
and B have same order say 
.
ii.
for 
If two matrices A and B are equal we write 
Example: 
About symmetric, skew symmetric, Hermitian, skew hermitian etc.
Will be updated soon………
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