1. Matrices and Determinants
UNIT 4
Matrices and Determinants
4.1 Matrices
4.2 Types of Matrices
4.3 Algebra of matrices
4.4 Determinant of matrices
4.5 Adjoint of matrix
4.6 Inverse of matrix
4.7 Homogeneous equation
4.8 Linear equation

2. Introduction
Matrix is a powerful tool in modern mathematics as it has wide applications. It can be used for practical business purposes hence occupies important place in business mathematics. The matrix form therefore suits very well for allocation of Expenses, budgeting for by-products etc.

3. 4.1 Matrices
A matrix consists of a rectangular presentation of symbols or numerical elements arranged systematically in rows and columns describing various aspects of a phenomenon inter- related in some manner.
A) Multivariable Data :
Statistics and multivariate analysis heavily rely on the use of matrix algebra. In simple words matrix is an arrangement of variables in Rows and columns.
Let us consider one example :
A company owns there factories ‘A’, ‘B’, ‘C’ which produce two products P and Q. So we have multivariate data regarding factory A producing product P and Q factory B producing product P and Q and factory C producing product P and Q. We can consider the above example as product P is produced in factory A, B and C and product Q is produced in A, B and C factories so to represent this multivariate data we can use matrix.

4. 4.1 Matrices For E.g. : A produces 4000 units of P and 5000 units of Q
B produces 3000 units of P and 2000 units of Q
C produces 1000 units of P and 5000 units of Q.
The above information can be represented in rows and columns as.
Factories
A B C P
Product Q
Each row represents the no. of units of particular product and each column represent the no. of units of different products by a particular factory.
The arrangement of this information is called as matrix
[ ]

5. 4.1 Matrices B) Definition of Matrix :
i) A rectangular arrangement of numbers in ‘m’ rows and ‘n’ columns is called a matrix of order ‘m x n’
ii) A matrix is a rectangular array of numbers arranged in rows and columns enclosed by a pair of brackets and subject to certain rules of presentation.
Sometimes a pair of brackets [ ], or a pair of double bars II II are used instead of a pair of parentheses ( ) :
e.g. or

6. 4.1 Matrices C) Order of Matrix :
Order of matrix is (m x n ) order where m represents no. of Rows and n represents no. of columns.
for e.g.
A =
Order of matrix is 3 x 3 3 rows and 3 colums
B =
is order of ( 2 x 3) matrix
2 Rows and 3 columns.

7. 4.1 Matrices
A matrix consists of a rectangular presentation of symbols or numerical elements arranged systematically in rows and columns describing various aspects of a phenomenon inter- related in some manner.
A) Multivariable Data :
Statistics and multivariate analysis heavily rely on the use of matrix algebra. In simple words matrix is an arrangement of variables in Rows and columns.
E.g. :
A produces 4000 units of P and 5000 units of Q
B produces 3000 units of P and 2000 units of Q
C produces 1000 units of P and 5000 units of Q.
The above information can be represented in rows and columns as.
Factories
A B C P
Product Q
Each row represents the no. of units of particular product and each column represent the no. of units of different products by a particular factory.
The arrangement of this information is called as matrix
[ ]

8. 4.1 Matrices B) Definition of Matrix :
i) A rectangular arrangement of numbers in ‘m’ rows and ‘n’ columns is called a matrix of order ‘m x n’
ii) A matrix is a rectangular array of numbers arranged in rows and columns enclosed by a pair of brackets and subject to certain rules of presentation.
Sometimes a pair of brackets [ ], or a pair of double bars II II are used instead of a pair of parentheses ( ) :
Example:
C) Order of Matrix :
Order of matrix is (m x n ) order where m represents no. of Rows and n represents no. of columns.
for e.g.
A =
Order of matrix is 3 x 3 3 rows and 3 columns

9. 4.1 Matrices D) Notations:
A matrix is usually denoted by a capital letter and its elements by corresponding small letters followed by two suffixes, the first one indicating the row and second one the column in which the element appears.
A rectangular arrangement of numbers in m rows and n columns is called a matrix of order m x n matrix can be written as.
A =
The entries in matrix are called as elements so the elements are a11, a amn

10. 4.2 Types of Matrices 1) Null Matrix :
If all elements of a matrix are zero the matrix is called a null matrix or zero matrix.
Example:
A =
In this all elements are zero.
2) Row Matrix :
A matrix in which there is only one row and n columns. It is (1 x n) order matrix and n matrix.
A = [ 5 7 6] (1 x 3) matrix
The above matrices are called as Row matrices as there is only one row in matrices.

11. 4.2 Types of Matrices 3) Column Matrix:
A matrix in which there is only one column and m Rows is called as column matrix. It is in the order of [ m x 1]
Example:
A = B =
2 x 1 matrix 3 x 1 matrix
4) Square Matrix :
A matrix in which the number of rows are equal to number of columns, is called as square matrix. Thus m x n matrix is a square matrix if m = n.
A = B =
2 x 2 matrix 3 x 3 matrix

12. 4.2 Types of Matrices 5) Diagonal Matrix :
A square matrix in which all elements except diagonal elements i.e. all non- diagonal elements are zero, it is called as diagonal matrix .
Example:
A = B =
6) Unit Matrix :
A diagonal matrix in which all Diagonal elements is unity i.e. ‘1’ (one) called as unit matrix or an Identity matrix. A unit matrix of order n is written as In.
I2 =

13. 4.2 Types of Matrices 7) Scalar Matrix :
A square matrix when given in the form of a scalar. multiplication to an Identity matrix is called a scalar matrix.
Example:
3I = = 3
8) Triangular Matrix :
i) Upper Triangular Matrix :
A square matrix A = (aij) m x n is called upper triangular matrix if aij = 0 for i > j
ii) Lower Triangular Matrix :
A square matrix A = (aij) m x n is called lower triangular matrix if if aij = 0 for i < j
A =

14. 4.2 Types of Matrices 9) Symmetric Matrices.
A symmetric matrix is a special kind of a square matrix A = aij for which aij for all i and j i.e. the (i, j) the element = (j, i) the element.
Example:

15. 4.3 Algebra of Matrices
1) Additions of Matrices / Subtraction of Matrices :
Matrices can be added or subtracted if and only if they are of same order.
Example:
A =
B =
Both matrices are of same order that is 2 x 2 order.
Addition of matrices
A + B =
Subtraction of matrices.
A - B =

16. 4.3 Algebra of Matrices 2) Multiplication of Matrix :
For multiplication, the number of columns in the 1st matrix must be equal to number of rows in the second matrix.
Example:
A =
B =
In this case (A x B) is not possible as No. of columns in 1st matrix A is 3 and No. of rows in matrix B is 2 i.e. 3 is not 2
But (B. A) is possible as No. of columns in matrix B is 2 and No. of rows in matrix A is 3. (B x A) can be made.

17. 4.4 Determinant of Matrix 1) Consider a square of order 2 : A =
Its Determinant is
2) Consider a square matrix of order 3 then Determinant is
[A] =
[A] = a11
= a11( a22 a33 - a32 a23 ) - a12 ( a21 a33 - a31 a23) + a13 ( a21 a32 - a31 a22)

18. 4.5 Adjoint of Matrix 1) Minor of Matrix Consider a Matrix of order 2
Minor of element of aij is denoted as Mij defined as sub-matrix deleting Ist row and its corresponding column.
Example:
M11 =
M11 = a22
Minor of a12 =
= a21
Minor of a21 =
= a12
Minor of a22 =
= a11

19. 4.5 Adjoint of Matrix 2) Co-factor Matrix :
Let A be any square matrix then ith row and jth column element is denoted by aij and cofactor of (i, j)th element is :
Co-factor of aij = (-1)i + j.| Mij | = Aij
Where mij is minor of a matrix obtained by deleting ith row and jth column.
Example:
A =
Co factor
C11 = (-1) 1+1 | 3 | = 3
C12 = (-1) 1+2 | 4 | = -4
C21 = (-1) 2+1 | 2 | = -2
C22 = (-1) 2+2 | 7 | = 7 =

20. 4.5 Adjoint of Matrix 3) Adjoint of Matrix:
Adjoint of matrix is defined as transpose of cofactor matrix.
Adj (A) = (Cofactor matrix)t
Transpose of matrix is done by changing rows into columns and columns into Rows.
For the above two examples adjoint are as under
i) Cofactor Matrix =
Adjoint (A) =
ii) adj (A) =

21. 4.6 Inverse of Matrix
Given any square matrix A whose determinant is not zero. If we can find another Matrix B such that AB = BA =1 then B is called as Inverse of A. It is denoted by A-1
Formula is
A-1 = .adj (A)
Example:
A =
| A | = = 7
Minors Cofactors
M11 = 2 C11 = 2
M12 = 1 C12 = -1
M21 = 3 C21 = -3
M22 = 5 C22 = 5
Cofactor Matrix is =
Adj (A) =
A-1 = .adj (A) =

22. 4.7 Homogeneous Equations
A system of linear equations represented by the matrix equation. AX = 0, Where A is the coefficient matrix X the matrix formed by the variables and 0, the null matrix is called as homogeneous system of linear equation.
A) An Expression of the form :
In which every term is of the nth degree, is called a homogeneous function of degree n. You recall that a linear differential equation,
was called homogeneous if g (x) = 0.
Homogeneous matrix equations have some special properties :
1) The matrix equation AX = 0 always has at least one solution, the zero solution X = 0 (Here 0 stands for a column vector all of whose entries are zero.)
2) If the column vectors X1 and X2 are two solutions to the matrix equation AX = 0
then so is any linear combination of them,

23. 4.8 Linear Equation
An equation with degree one i.e. maximum power of variable is one. Such equation is called as linear equation.
Example:
i) a x = b Linear equation with one variable
ii) a x + by = c
A linear equation with two variable where a, b, and c are constant, x and y are variables.
iii) Linear equation with 3 variables
Where a, b, c, and z are constants.
Linear Equations :
A linear equation looks like any other equation. It is made up of two expressions set equal to each other. A linear equation is special because:
It has one or two variables.
2) No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.

24. 4.8 Linear Equation
3) When you find pairs of values that make the linear equation true and plot those pairs on a co-ordinate grid, all of the points for any one equation lie on the same line. Linear equations graph as straight lines.
4) Variables cannot have exponents (or powers).
5) Variables cannot multiply or divide each other.
6) Variables cannot be found under a root sign or square root sign (sqrt).