## 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 :**

## 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

**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.

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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