## Introduction

The determinant of a matrix is a number that associated with the square matrix. This number may be positive, negative or zero.

The determinant of the matrix A is denoted by det A or |A| or D

For 2 by 2 matrix-

Determinant will be

For 3 by 3 matrix-

Determinant will be

## Solved examples

**Example: If then find |A|.**

Sol.

Sol.

As we know that-

Then

**Example: Find out the determinant of the following matrix A.**

Sol.

By the rule of determinants-

**Example: expand the determinant: **

Sol. As we know

Then

**Minor**

The minor of an element is define as determinant obtained by deleting the row and column containing the element.

In a determinant,

The minor of a1, b1 and c1 are given by

## Cofactor

Cofactors can be defined as follows,

Where r is the number of rows and c is the number of columns.

**Example: Find the minors and cofactors of the first row of the determinant.**

Sol. (1) The minor of element 2 will be,

Delete the corresponding row and column of element 2,

We get,

Which is equivalent to, 1 × 7 – 0 × 2 = 7 – 0 = 7

Similarly the minor of element 3 will be,

4× 7 – 0× 6 = 28 – 0 = 28

Minor of element 5,

4 × 2 – 1× 6 = 8 – 6 = 2

The cofactors of 2, 3 and 5 will be,

**Properties of determinants**

(1) If the rows are interchanged into columns or columns into rows then the value of determinants does not change.

Let us consider the following determinant:

(2) The sign of the value of determinant changes when two rows or two columns are interchanged.

Interchange the first two rows of the following, we get

(3) If two rows or two columns are identical the the value of determinant will be zero.

Let, the determinant has first two identical rows,

As we know that if we interchange the first two rows then the sign of the value of the determinant will be changed, so that

hence proved.

(4) if the element of any row of a determinant be each multiplied by the same number then the determinant multiplied by the same number,

**Example: Show that,**

Sol. Applying

We get

**Example: Solve-**

Sol:

Given

We get

**Applications of determinants**

Determinants have various applications such as finding the area and condition of collinearity.

**Area of triangles-**

Suppose the three vertices of a triangle are respectively, then we know that the area of the triangle is given by-

This is how we can find the area of the triangle.

**Condition of collinearity-**

Let there are three points

Then these three points will be collinear if –

**Example: Show that the points given below are collinear-**

(11, 7), (5, 5) and (-1, 3)

Sol:

First we need to find the area of these points and if the area is zero then we can say that these are collinear points-

So that-

We know that area enclosed by three points-

So that these points are collinear.

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