Business Mathematics | Unit 1 Matrix and Determinants | B.COM 4th Sem Gauhati University

Business Mathematics Notes for B. Com 6th Sem

Business Mathematics Notes for B. Com 4th Sem

Gauhati University
 Important Questions

(Also Useful for Dibrugarh University and Assam University, Rabindra Nath Tegore University)

Unit 1 : MATRICES AND DETERMINANTS

Determinants

Define determinant.

Ans: An arrangement of numbers along horizontal lines (called rows) and vertical lines called columns enclosed by two vertical lines is said to be a determinant. There are equal number of rows and columns in a determinant. It is denoted by ∆ or A. 



Minor and Co factor:


The minor of an element aij is the determinant obtained by delating the row and column in which aij exists (i.e. ith row and jth column) and denoted by M




Co-factor : If we apply the appropriate sign to the minor of an element we have its co-factors. The co-factor of aij is denoted by Aij. Relation between co-factor and minor of an element aij is Aij=(-1)i+j Mij


Properties of determinant :


Properties of determinant, business mathematic Guwahati University properties of determinant b.com 4th sem
Property of determinants for b.com 4th sem business mathematics Guwahati University

Business Mathematics
Business Mathematics

Business Mathematics


Solving equations using determinants (Cramer's Rule) :

Cramer's rule for three equations in three variables. Consider the system of three linear equation in three variables x, y, z.


Cramer’s Rule Formula

Consider a system of linear equations with n variables x₁, x₂, x₃, …, xₙ written in the matrix form AX = B.

Here,

A = Coefficient matrix (must be a square matrix)

X = Column matrix with variables

B = Column matrix with the constants (which are on the right side of the equations)

Now, we have to find the determinants as:

D = |A|, Dx1, Dx2, Dx3,…, Dxn

Here, Dxi for i = 1, 2, 3,…, n is the same determinant as D such that the column is replaced with B.

Thus,

x1 = Dx1/D; x2 = Dx2/D; x3 = Dx3/D; ….; xn = Dxn/D {where D is not equal to 0}


Cramer’s Rule 2×2


Cramer’s Rule 2×2



Question:

Solve the following system of equations using Cramer’s rule:

2x – y = 5

x + y = 4

Solution:

Given,

2x – y = 5

x + y = 4

Let us write these equations in the form AX = B.


Question:  Solve the following system of equations using Cramer’s rule:  2x – y = 5  x + y = 4  Solution:  Given,  2x – y = 5  x + y = 4  Let us write these equations in the form AX = B.


Cramer’s Rule 3×3

To find the Cramer’s rule formula for a 3×3 matrix, we need to consider the system of 3 equations with three variables.

Consider:

a1x + b1y + c1z = d1

a2x + b2y + c2z = d2

a3x + b3y + c3z = d3

Let us write these equations in the form AX = B.


Cramer’s Rule 3×3 To find the Cramer’s rule formula for a 3×3 matrix, we need to consider the system of 3 equations with three variables.  Consider:  a1x + b1y + c1z = d1  a2x + b2y + c2z = d2  a3x + b3y + c3z = d3  Let us write these equations in the form AX = B.

Therefore, x = Dx/D, y = Dy/D, z = Dz/D; D ≠ 0

Go through the example given below to learn how to solve Cramer’s rule for the 3×3 matrix.


Cramer’s Rule Example – 3×3

Question:

Solve the following system of equations using Cramer’s rule:

x + y + z =6

y + 3z = 11

x + z =2y or x – 2y + z = 0

Solution:



Matrices


Q1. Define Matrices. 

Ans: m x n real numbers arranged in m rows and n columns and enclosed by a pair of brackets [ ] or () is called an mxn matrix (read as m by n) matrix.


If m=n, i.e. if the number of rows is equal to the number of columns the number of each being n then the matrix is called a square matrix.


Q2. Type of Matrices


(i) Row Matrix: A matrix containing only one row is called a row matrix


Ex. [ 1  6  8 ] 1x3


(ii) Column Matrix: A matrix containing only one column is called a column Matrix.




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