Assignment of matrix-algebra

1. Name : Prakhar Pathak
Class : B.Com
Roll No. : 2101214

2. MATRIX-
ALGEBRA

3. Algebra of Matrices is the branch of mathematics, which
deals with the vector spaces between different dimensions.
The innovation of matrix algebra came into existence
because of n-dimensional planes present in our coordinate
space.
Algebra of matrix involves the operation of matrices, such
as Addition, subtraction, multiplication etc.

4. A matrix is an ordered rectangular array of
numbers, arranged in rows and columns.
The size or order of a matrix is described by
its number of rows and the number of
columns.
If a matrix, A, has m rows and n columns
then A is described as an mxn matrix.

5. The numbers in a matrix are called its elements. The
element in the Ith row and Jth column of a matrix is
generally denoted by all. A matrix with m rows and n
columns is written (aij)mn or (aij).

6. Row MatrixA matrix with just one row is
called a row matrix (or row vector).A =
a1,a2..., an = αj (1xn)
A matrix with just one column is called a
column matrix.

8. Two matrices which have the Same
number of rows and columns are said
to be matrices of the same order.

9. Two matrices A = (a) and B = (b) are said to be
equal if, and only if, each element a, of A is equal to
the corresponding element b,, of B.
In symbolic form this reads:A=B:- aij = bij for all i
and j
From this it follows that equal matrices are of the
same order but matrices of the same order are not
necessarily equal.

10. Any matrix, all of whose elements are zero, is
called a null or zero matrix and is denoted by
0.

11. a square matrix in which all the elements of
the principal diagonal are ones and all other
elements are zeros. The effect of multiplying
a given matrix by an identity matrix is to
leave the given matrix unchanged.

12. The transpose of a matrix is found by interchanging
its rows into columns or columns into rows. The
transpose of the matrix is denoted by using the letter
“T” in the superscript of the given matrix. For
example, if “A” is the given matrix, then the transpose
of the matrix is represented by A’ or AT.

13.  Addition of matrix is the basic operation performed, to add two or more
matrices. Matrix addition is possible only if the order of the given matrices
are the same. By order we mean, the number of rows and columns are the
same for the matrices. Hence, we can add the corresponding elements of
the matrices.
 The addition of algebraic expressions can only be done with the
corresponding like terms, similarly the addition of two matrices can be
done by addition of corresponding terms in the matrix.
 There are basically two criteria that define the addition of a matrix. They
are as follows:
 Consider two matrices A & B. These matrices can be added if (if and only
if) the order of the matrices are equal, i.e. the two matrices have the same
number of rows and columns. For example, say matrix A is of the order
3×4, then the matrix B can be added to matrix A if the order of B is
also3×4
 The addition of matrices is not defined for matrices of different sizes.

14. For Example :

15. Matrix subtraction is exactly the same as matrix addition.
All the constraints valid for addition are also valid for
matrix subtraction. Matrix subtraction can only be done
when the two matrices are of the same size. Subtraction
cannot be defined for matrices of different sizes.
In other words, it can be said that matrix subtraction is an
addition of the inverse of a matrix to the given matrix

16. For Example :

17.  Matrix multiplication, also known as matrix product and the
multiplication of two matrices, produces a single matrix. It is a type of
binary operation .
 If A and B are the two matrices, then the product of the two matrices A
and B are denoted by:
 X = AB
 Hence, the product of two matrices is the dot product of the two matrices.
 Matrix can be Multiplied two ways,
 (i) Scalar Multiplication
 (ii) Multiplication with another matrix

18. Multiplication of an integer with a matrix is simply scalar
multiplication.
We know that a matrix is an array of numbers. It consists
of rows and columns. It involves multiplying a scalar
quantity to the matrix. Every element inside the matrix is
to be multiplied by the scalar quantity to form a new
matrix.
A scalar quantity is any real number starting from one till
infinity.

19. For Example :

20. To perform multiplication of two matrices, we should make
sure that the number of columns in the 1st matrix is equal
to the rows in the 2nd matrix. Therefore, the resulting
matrix product will have a number of rows of the 1st
matrix and a number of columns of the 2nd matrix. The
order of the resulting matrix is the matrix multiplication
order.

21. For Example :