Determinants - Why Study Properties Of Determinants

Determinants - Why study properties of determinants

Determinants are important mathematical concepts used in various fields such as physics, engineering, and economics. Understanding their properties is crucial for solving a wide range of problems.
Some reasons why we study properties of determinants include:
- Evaluating solutions of linear systems of equations
- Calculating the area of geometric figures
- Determining whether a matrix is invertible
- Solving systems of linear equations
- Finding the inverse of a matrix
By studying the properties of determinants, we gain a deeper understanding of matrices and their applications in real-life scenarios.
Properties of determinants enable us to perform various matrix operations efficiently and accurately.
It helps us simplify complex calculations and provide elegant solutions to complex problems.
Determinants play a crucial role in linear algebra and in understanding the behavior of various mathematical systems.
Let’s explore some important properties of determinants that will enhance our problem-solving skills in linear algebra.
Understanding the properties of determinants will serve as a foundation for more advanced mathematical concepts and applications.
In this lecture, we will cover some essential properties of determinants, their proofs, and how to apply them in different scenarios.

Properties of Determinants

Property 1: If we multiply a row (or column) of a determinant by a scalar k, the value of the determinant is multiplied by k.
Property 2: If two rows (or columns) of a determinant are interchanged, the sign of the determinant is changed.
Property 3: If we add a multiple of one row (or column) of a determinant to another row (or column), the value of the determinant remains unchanged.
Property 4: If a determinant has two identical rows (or columns), the value of the determinant is zero.
Property 5: If all elements of a row (or column) of a determinant are zero, then the value of the determinant is zero.
Property 6: If a determinant has all its elements in any one row (or column) equal to zero, then the value of the determinant is zero.
Property 7: If a determinant is multiplied by a constant factor and then added to another determinant of the same order, the value of the resulting determinant is equal to the sum of the values of the two determinants.
Property 8: If a multiple of a row (or column) is added to another row (or column) of a determinant, then the value of the determinant remains unchanged.
Property 9: If a determinant has two identical rows (or columns), then the value of the determinant is zero.
Property 10: The value of the determinant of a square matrix remains unchanged by taking the transpose of the matrix.
Property 11: If a determinant has a row (or column) consisting entirely of zeros, then the value of the determinant is zero.
Property 12: If two rows (or columns) of a determinant are proportional to each other, then the value of the determinant is zero.
Property 13: If any two rows (or columns) of a determinant are interchanged, the value of the determinant changes sign.
Property 14: The value of the determinant is unchanged if any two rows (or columns) of the determinant are interchanged.
Property 15: The value of the determinant remains unchanged if the elements of a row (or column) are multiplied by a non-zero scalar k.
Property 16: If the rows (or columns) of a determinant are interchanged, and each element of one row (or column) is multiplied by k, the value of the determinant remains unchanged.
Property 17: If any two rows (or columns) of a determinant are identical, then the value of the determinant is zero.
Property 18: The value of the determinant remains unchanged if each element of a row (or column) is multiplied by a constant and added to the corresponding element in another row (or column).
Property 19: The value of the determinant remains unchanged if the elements of a row (or column) are multiplied by a constant k and added to the corresponding elements of another row (or column).
Property 20: The value of the determinant remains unchanged if all the elements of a row (or column) are multiplied by a constant k and the resulting determinant is multiplied by 1/k.

Proof of Property 1: If we multiply a row (or column) of a determinant by a scalar k, the value of the determinant is multiplied by k.

Let’s consider a determinant A of order n.
Suppose we multiply the elements of the ith row (or column) of A by a scalar k, denoted by Ak.
The new determinant will be represented by Ak.
We can write Ak as Ak = [Cij], where Cij = k * Aij, for i = 1 to n and j = 1 to n.
By expanding the determinant Ak, we can observe that every element of the ith row (or column) in the original determinant A gets multiplied by k.
Therefore, Ak = k * A.
Hence, the value of the determinant is multiplied by k.

Proof of Property 2: If two rows (or columns) of a determinant are interchanged, the sign of the determinant is changed.

Let’s consider a determinant A of order n.
Suppose we interchange the ith and jth rows (or columns) of A, denoted by Ai and Aj.
The new determinant will be represented by AiAj.
By expanding the determinant AiAj, we can observe that the element in the ith row (or column) in the original determinant A becomes the element in the jth row (or column) in AiAj.
Similarly, the element in the jth row (or column) in the original determinant A becomes the element in the ith row (or column) in AiAj.
Since the elements of Ai and Aj have been interchanged, the sign of each term in the new determinant is changed.
Therefore, AjAi = -A.
Hence, the sign of the determinant is changed.

Proof of Property 3: If we add a multiple of one row (or column) of a determinant to another row (or column), the value of the determinant remains unchanged.

Let’s consider a determinant A of order n.
Suppose we add p times the ith row (or column) of A to the jth row (or column), denoted by Aj = Ai + pAi.
The new determinant will be represented by Aj.
By expanding the determinant Aj, we can observe that every term in Aj consists of the sum of the corresponding terms in Ai and pAi.
Since the terms in Ai and pAi have a common factor of (1 + p), we can factor out (1 + p) from each term in Aj and obtain Aj = (1 + p)Ai.
Therefore, Aj is a scalar multiple of Ai.
The value of a determinant is zero if two rows (or columns) are proportional.
Since Aj is proportional to Ai, the value of the determinant remains unchanged.
Hence, the value of the determinant remains unchanged.

Proof of Property 4: If a determinant has two identical rows (or columns), the value of the determinant is zero.

Let’s consider a determinant A of order n.
Suppose A has two identical rows (or columns) denoted by Ai and Aj.
If Ai and Aj are identical, then subtracting them will result in a row (or column) of zeros, denoted by Ak = Ai - Aj.
The new determinant will be represented by Ak.
By expanding the determinant Ak, we can observe that every term in Ak consists of the difference of the corresponding terms in Ai and Aj, which is zero.
Hence, each term in Ak is zero, and therefore, the value of the determinant is zero.
Therefore, if a determinant has two identical rows (or columns), the determinant is zero.

Proof of Property 5: If all elements of a row (or column) of a determinant are zero, then the value of the determinant is zero.

Let’s consider a determinant A of order n.
Suppose all elements of the jth row (or column) of A are zero.
We can write this determinant as A = [Aij], where Aij = 0 for all i = 1 to n.
By expanding the determinant A, we can observe that every term in the sum of the determinant consists of the product of the corresponding elements in each row (or column).
Since all elements of the jth row (or column) are zero, every term in the determinant A will also be zero.
Therefore, the value of the determinant is zero if all elements of a row (or column) are zero.

Example of Property 1:

Let’s consider the determinant A = |2 4|, and we multiply each element of the second row by 3: |1 5|
The new determinant will be represented by A’ = |2 4|, where every element of the second row is multiplied by 3: |3 15|
Calculating the determinant, we have |2 4| = (2 * 15) - (4 * 3) = 30 - 12 = 18. |3 15|

Example of Property 2:

Let’s consider the determinant A = |2 3|, and we interchange the first and second rows: |5 6|
The new determinant will be represented by A’ = |5 6|, where the first row and second row are interchanged: |2 3|
Calculating the determinant, we have |5 6| = (5 * 3) - (6 * 2) = 15 - 12 = 3. |2 3|

Example of Property 3:

Let’s consider the determinant A = |2 3|, and we add 4 times the first row to the second row: |1 2|
The new determinant will be represented by A’ = |2 3|, where 4 times the first row is added to the second row: |9 14|
Calculating the determinant, we have |2 3| = (2 * 14) - (3 * 9) = 28 - 27 = 1. |9 14|

Example of Property 4:

Let’s consider the determinant A = |2 4|, where the first and second rows are identical: |2 4|
Since the determinant has two identical rows, the value of the determinant is zero.

Example of Property 5:

Let’s consider the determinant A = |0 0|, where all elements of the first row are zero: |1 2|
Since all elements of the first row are zero, the value of the determinant is zero.