Table of Contents
Determinants and Its Properties
Determinant is a special function of a square matrix. It is a scalar quantity that is defined as the product of the diagonal elements of the matrix, raised to the appropriate power.
Some of the important properties of determinant are:
1. Determinant is a function of the matrix and does not depend on the order of the matrix.
2. If the determinant of a matrix is zero, the matrix is said to be singular and has no inverse.
3. If the determinant of a matrix is not zero, the matrix is said to be nonsingular and has an inverse.
4. The determinant of a product of two matrices is the product of the determinants of the individual matrices.
5. The determinant of a sum of two matrices is the sum of the determinants of the individual matrices.
What is known as Determinants?
A determinant is a mathematical function that calculates the size of a matrix. It is denoted with a small letter “d” followed by the square brackets enclosing the matrix. The determinant of a matrix is calculated by multiplying each element of the matrix by its corresponding determinant coefficient and then summing the results.
How is a Determinant different from a Matrix?
A determinant is a function that takes a square matrix as input and outputs a scalar. A matrix is a two-dimensional array of numbers.
Properties of Determinants – Explanation, Important Properties, and FAQs
What is a determinant?
A determinant is a function that assigns a scalar value to every row and column of a square matrix.
What are the important properties of determinants?
The important properties of determinants are:
1. The determinant of a matrix is not affected by the order of the matrix’s elements.
2. The determinant of a matrix is equal to the product of the determinants of the matrix’s individual rows or columns.
3. The determinant of a matrix is zero if and only if the matrix is singular (has no inverse).
4. The determinant of a matrix is a function of the matrix’s elements and is not dependent on the matrix’s order.
Important Properties of Determinants
A determinant is a square matrix that is calculated by taking the determinant of each individual matrix element and multiplying the results together. The determinant of a matrix is a scalar value, and it is always positive.
The determinant of a matrix is also unique, meaning that it is the only scalar value that can be calculated from the matrix. Additionally, the determinant is unaffected by the addition or subtraction of a row or column from the matrix.
1. Reflection Property
The reflection property is a Boolean property that returns true if an object is a reflection of the given object.
2. All- Zero Property
An element is said to be all-zero if its value is all zeros. For example, the all-zero property of the number 0 is that it is equal to 0.
3. Even-Odd Property
An element is said to have the even-odd property if the remainder when it is divided by 2 is even or odd. For example, the number 5 has the even-odd property because the remainder when it is divided by 2 is 1, which is odd.
3. Proportionality (Repetition Property)
If a set of objects is repeated n times, then the total size of the set is n times the size of the original set.
6. Scalar Multiple Property
7. Vector Property
8. Matrix Property
7. Sum Property
The sum property states that the sum of two or more numbers is equal to the sum of their individual values.
For example, the sum of 4 and 5 is 9.
8. Triangle Property
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
This theorem is often called the Triangle Sum Theorem.
9. Determinant of Cofactor Matrix
The determinant of a cofactor matrix is the determinant of the original matrix multiplied by the determinant of the transpose of the original matrix.
Examples Problems on Properties of Determinants
1. Find the determinant of the matrix:
2. Find the determinant of the matrix:
3. Find the determinant of the matrix:
4. Find the determinant of the matrix:
5. Find the determinant of the matrix:
6. Find the determinant of the matrix:
7. Find the determinant of the matrix:
8. Find the determinant of the matrix:
9. Find the determinant of the matrix:
10. Find the determinant of the matrix:
1. Using Properties of Determinants, Prove That
|A| = det(A)
The property of determinants states that the determinant of a matrix is equal to the product of the diagonal elements of the matrix. Therefore, |A| = det(A) = ad – bc.
2. Using Properties of Determinants, Prove That
| det ( A ) | = | det ( A T ) |
The determinant of a matrix is a scalar value that is independent of the order of the matrix. The determinant of a matrix is also equal to the determinant of its transpose.
