Determinants - Examples On Determinants

Determinants - Examples on Determinants

Today’s Lesson: Determinants
Examples of solving determinants
Understanding the properties of determinants
How to find the value of a determinant
Practical applications of determinants in real-life problems

What is a Determinant?

A determinant is a mathematical value that can be calculated from a square matrix.
It provides important information about the matrix, such as whether it is invertible or singular.
The determinant of a matrix is denoted by “det(A)” or “|A|”.

Properties of Determinants

The determinant of the identity matrix is always 1: det(I) = 1.
If two rows or two columns of a matrix are interchanged, the sign of the determinant changes.
Multiplying a row or column by a constant multiplies the determinant by that constant.
If two rows or two columns of a matrix are identical, the determinant of that matrix is zero.

How to Find the Value of a Determinant?

For a 2x2 matrix:
- Given A = | a b | | c d |
- The determinant of A is ad - bc.

Example: Finding the Determinant of a 2x2 Matrix

Given A = | 2 3 | | 4 5 |
To find the determinant of A, we multiply the elements along the main diagonal and subtract the product of the elements along the other diagonal.
Thus, det(A) = (2 * 5) - (3 * 4) = 10 - 12 = -2

Solving Determinants for a 3x3 Matrix

Given A = | a b c | | d e f | | g h i |
The determinant of a 3x3 matrix can be found using the formula:
- det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Example: Finding the Determinant of a 3x3 Matrix

Given A = | 1 2 3 | | 4 5 6 | | 7 8 9 |
Applying the formula,
- det(A) = 1(5 * 9 - 6 * 8) - 2(4 * 9 - 6 * 7) + 3(4 * 8 - 5 * 7) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0

The Cofactor Expansion Method

The cofactor expansion method is another way to find the determinant of a matrix.
It involves expanding the determinant along a row or column by using cofactors.
Cofactors are obtained by determining the determinants of submatrices formed by removing a row and column from the original matrix.

Example: Using the Cofactor Expansion Method

Given A = | 1 2 3 | | 4 5 6 | | 7 8 9 |
Expanding along the first row, we have:
- det(A) = 1C11 - 2C12 + 3*C13
- C11, C12, and C13 are the cofactors of the elements 1, 2, and 3 respectively.

Recap: Determinants

The determinant of a matrix provides important information about the matrix.
Determinants can be found for 2x2 and 3x3 matrices using different methods.
The properties of determinants help in simplifying calculations and solving real-life problems.
Practice solving more examples to reinforce your understanding.

Determinants - Examples on Determinants

In this section, we will solve some examples to understand the concept of determinants better.
We will practice calculating determinants for matrices of different sizes.
These examples will help reinforce our understanding and prepare for solving more complex problems.

Example 1: Finding the Determinant of a 2x2 Matrix

Given A = | 3 2 | | 1 4 |
To find the determinant of A, we multiply the elements along the main diagonal and subtract the product of the elements along the other diagonal.
Thus, det(A) = (3 * 4) - (2 * 1) = 12 - 2 = 10

Example 2: Finding the Determinant of a 3x3 Matrix

Given A = | 2 1 3 | | 0 5 2 | | 1 4 6 |
Applying the formula,
- det(A) = 2(5 * 6 - 2 * 4) - 1(0 * 6 - 2 * 1) + 3(0 * 4 - 5 * 1) = 2(30 - 8) - 1(0 - 2) + 3(0 - 5) = 2(22) - 1(-2) - 3(-5) = 44 + 2 + 15 = 61

Example 3: Finding the Determinant of a 4x4 Matrix

Given A = | 1 2 3 4 | | 0 1 2 3 | | 3 4 5 6 | | 2 3 0 1 |
Using the cofactor expansion method, we can expand along the first row:
- det(A) = 1C11 - 2C12 + 3C13 - 4C14
- C11, C12, C13, and C14 are the cofactors of the elements 1, 2, 3, and 4 respectively.

Example 4: Determinant of a Square Matrix

Given A = | a b c d | | e f g h | | i j k l | | m n o p |
The determinant of a square matrix A is given by the formula:
- det(A) = a(fkp - gjo - hlp + gln + hkm - fln) - b(ekp - gip - hlp + gin + hmo - ein) + c(egp - fip - hio + fin + hmo - gjm) - d(egn - fjm - gjo + fkn + hkm - ejn)

Example 5: Determinant of a 5x5 Matrix

Given A = | a b c d e | | f g h i j | | k l m n o | | p q r s t | | u v w x y |
Using the cofactor expansion method, we can expand along the first row:
- det(A) = a * C11 - b * C12 + c * C13 - d * C14 + e * C15
- C11, C12, C13, C14, and C15 are the cofactors of the elements a, b, c, d, and e respectively.

Applications of Determinants

Determinants have wide applications in various fields, including:
- Solving systems of linear equations
- Calculating the area of polygons
- Finding the volume of parallelepipeds
- Inverse matrices and matrix transformations
- Determining the existence of solutions to equations

Example with Application: Solving Systems of Equations

Given the system of equations:
- 2x + 3y = 7
- 4x - y = 10
We can write the system in matrix form as AX = B, where A is the coefficient matrix, X is the variable matrix, and B is the matrix of constants.
To solve for X, we can use the equation X = A^(-1) * B, where A^(-1) is the inverse of matrix A.

Example with Application: Calculating the Area of a Triangle

Given the vertices of a triangle as coordinates (x1, y1), (x2, y2), and (x3, y3):
- The area of the triangle can be calculated using the formula:
  - Area = 1/2 * | x1 y1 1 | | x2 y2 1 | | x3 y3 1 |
- The absolute value of the determinant provides the area of the triangle.

Summary

In this lecture, we learned about determinants and their properties.
We solved examples for finding determinants of matrices of different sizes.
Determinants have various applications, such as solving systems of equations and calculating areas of polygons.
It is important to practice solving more examples and understanding the concepts thoroughly.

Solving Systems of Equations using Determinants

When solving a system of linear equations using determinants:
- Create the coefficient matrix A and the matrix of constants B.
- Find the determinant of A, |A|.
- If |A| is non-zero, the system has a unique solution.
- If |A| is zero, the system may have either no solution or infinitely many solutions.

Example: Solving a System of Equations

Given the system of equations:
- 2x + y = 5
- 3x - 2y = 1
Writing the system in matrix form: AX = B.
Coefficient matrix A:
- | 2 1 | | 3 -2 |
Matrix of constants B:
- | 5 | | 1 |
Calculating the determinant of A, |A| = (2 * -2) - (1 * 3) = -4 - 3 = -7.

Example: (continued)

Since |A| ≠ 0, the system has a unique solution.
The inverse of matrix A, A^(-1), can be found using the formula A^(-1) = (1/|A|) * adj(A), where adj(A) is the adjugate matrix.
A^(-1) = (1/-7) * |-2 1 | |-3 2 | = |-2/7 -1/7 | | 3/7 -2/7 |

Example: (continued)

Multiplying A^(-1) with B, we can find the solution X: X = A^(-1) * B.
A^(-1) * B = |-2/7 -1/7 | * | 5 | | 3/7 -2/7 | | 1 | = (-2/7 * 5) + (-1/7 * 1) (3/7 * 5) + (-2/7 * 1) = -10/7 - 1/7 15/7 - 2/7 = -11/7 + 13/7 = 2/7

Calculating the Area of a Polygon using Determinants

To calculate the area of a polygon using determinants:
- Create a matrix A, where each row represents the coordinates of a vertex of the polygon.
- Take the absolute value of the determinant of A, |A|.
- Divide |A| by 2 to get the area of the polygon.

Example: Calculating the Area of a Quadrilateral

Given the vertices of a quadrilateral as coordinates (x1, y1), (x2, y2), (x3, y3), and (x4, y4):
The area of the quadrilateral can be calculated using the formula:
- Area = 1/2 * | x1 y1 1 | | x2 y2 1 | | x3 y3 1 | | x4 y4 1 |
Taking the absolute value of the determinant provides the area of the quadrilateral.

Example: (continued)

Given the vertices of a quadrilateral as:
- (1, 2), (3, 5), (7, 6), (9, 1)
Using the formula:
- Area = 1/2 * | 1 2 1 | | 3 5 1 | | 7 6 1 | | 9 1 1 |
Expanding the determinant:
- Area = 1/2 * (1 * 5 - 2 * 1 + 3 * 6 - 5 * 9 + 7 * 1 - 6 * 9)
- Area = 1/2 * (5 - 2 + 18 - 45 + 7 - 54)
- Area = 1/2 * (-71)
- Area = -71/2

Finding the Volume of a Parallelepiped using Determinants

To find the volume of a parallelepiped using determinants:
- Create a matrix A, where each row represents the vectors representing the edges of the parallelepiped.
- Take the absolute value of the determinant of A, |A|, to get the volume of the parallelepiped.

Example: Finding the Volume of a Parallelepiped

Given the vectors representing the edges of a parallelepiped as:
- u = (1, 2, 3)
- v = (4, 5, 6)
- w = (7, 8, 9)
Creating the matrix A:
- | 1 2 3 | | 4 5 6 | | 7 8 9 |
Taking the absolute value of the determinant:
- Volume = |A| = | 1 2 3 | | 4 5 6 | | 7 8 9 |

Example: (continued)

Expanding the determinant:
- Volume = |A| = (1 * 5 * 9 + 2 * 6 * 7 + 3 * 4 * 8) - (3 * 5 * 7 + 2 * 4 * 9 + 1 * 6 * 8)
- Volume = |A| = (45 + 84 + 96) - (105 + 72 + 48)
- Volume = |A| = 225 - 225
- Volume = 0