Determination and Comparison

Determinants are scalar quantities that are obtained by adding the elements of a square matrix and their cofactors. To find out the adjoint inverse of a matrix, determinants are used. The determinants are functions of stretching out and shrinking back in the matrix. So, these take up the squared matrix as inputs and return a single number as the output.

Definition 

Determinant is a function.

Determinant: {squared matrix} → R

Determinants should fulfill the given properties-

• Any change in the row does not alter the value of det A

• Increasing the row of A by a scalar quantity c, multiplies the determinant by c

• If exchanging the two rows in a matrix, the determinant is multiplied by -1

• Determinant of an identity matrix In, is 1

For example- if we compute 

• The last row of the matrix is called the identity matrix, so its determinant is equal to 1

• The second last row was a replacement row, so the second final matrix has a determinant equal to 1

• The second step has a row-scaling by -1/7, so the determinant has to be -7

•  The first step in the row reduction is a row exchange, so the determinant of the first matrix is the negative of the second determinant

• The overall determinant of the matrix is 7

A general method for calculating the determinant is as follows-

Let A be a square matrix and suppose that some row row operations are being done on A to get Matrix B in row echelon form. So, 

Det (A) = (-1)r . (Product of diagonal entries of B) / Product of scaling factors used

r is the number of rows swapped.

The determinant of A is the product of the diagonal entries of the row echelon for B, times a factor of +-1 coming from the number of row swaps you made, divided by the product of the scaling factors used in the row reduction. 

Properties of Determinants

1. Reflection property- The determinant is not changeable if its rows and columns are interchanged. This is called the property of reflection.

2. All zero property- if all the items of a row/ column are equal to zero, then the determinant is zero. 

3. Repetition property- if all the elements of a row are in ratio with all the other elements of other rows or if all the elements of columns are similar to the other elements of a column, then the determinant are zero.

4. Switching property- if any two rows or columns are interchanged, the determinant has a changed sign.

5. Scalar multiple property- if the items of a row or column are multiplied by a constant which is non-zero, then the determinant gets multiplied by the same constant.

6. Sum property- 

7. Property of invariance

8. Factor property- if any determinant, let’s say A becomes 0, when we put, x= ∞, then, x-∞ is a symbol of A

9. Triangle Property- if all the elements of a determinant are above or below the main diagonal consisting of zeroes, then the determinant is equal to the product of diagonal elements. 

Comparison of Numbers 

Comparison of numbers is a process in which it is determined whether two or more numbers are equal to or greater than one another or lesser than one another. 

The basic symbols used in the comparison of numbers are-

• Greater than (>)

• Lesser than( <)

• Equal to( =)

Rules for comparison of numbers

There are two main rules for comparison of numbers:-

1. Numbers having different digits- When comparing numbers, the number having more digits is larger than the number having lesser number of digits, for example- Among the numbers- 8788, 566, 48, 9; 8788 is the largest number as it contains the most number of digits. Therefore, while comparing, the number having the greeter number of digits would always be the greater one. 

2. Numbers having the same number of digits – when comparing the number which have the same number of digits, we start from the extreme leftmost digits of the number. Therefore, the number with the greater extreme leftmost digit is always the greater one. 

3. For example- among the numbers 456, 768, and 987; 987 is the greatest as it has the greatest extreme leftmost digit.In the case of a number having the same number of extreme leftmost digits, the second number from the left is observed and the number having the greater second leftmost digit is the greater one. For example, out of 654 and 612; 645 is the greater one. 

Comparison of Integers

1. Integers are combinations of positive and negative numbers along with zero.  So, on moving towards the right, on the number line, the numbers are increasing and while moving towards the left, the values are decreasing. 

2. The numbers which lie on the extreme right is the greatest, and the number which lies on the extreme left is the smallest.

3. All positive numbers are greater than all negative numbers

4. Any negative integer is lesser than positive integers.

5. Zero is greater than all negative integers.

6. Zero is the least of all positive integers.

Comparison of Decimal Numbers

Decimal numbers have the whole number part and the decimal number part. So, the decimal fraction with a greater whole number part is always the greater one. For example- in comparing 2.34 and 1.23, 2.34 is the larger number as it has the larger whole number part.