Determinants - Determinants And Area Of Triangles

Determinants - Determinants and Area of triangles

The determinant of a 2x2 matrix [a b; c d] is given by ad - bc
The determinant of a 3x3 matrix [a b c; d e f; g h i] is given by a(ei - fh) - b(di - fg) + c(dh - eg)
Determinants can be used to find the area of a triangle
The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Example: Determinant of a 2x2 matrix

Let’s calculate the determinant of the matrix [2 3; 4 1].

Determinant = (2 * 1) - (3 * 4)
Determinant = 2 - 12
Determinant = -10

Example: Determinant of a 3x3 matrix

Let’s calculate the determinant of the matrix [1 2 3; 4 5 6; 7 8 9].

Determinant = 1(5 * 9 - 6 * 8) - 2(4 * 9 - 6 * 7) + 3(4 * 8 - 5 * 7)
Determinant = 1(45 - 48) - 2(36 - 42) + 3(32 - 35)
Determinant = 1(-3) - 2(-6) + 3(-3)
Determinant = -3 + 12 - 9
Determinant = 0

Determinants and Area of Triangles

The determinant of a 2x2 matrix can be used to find the area of a triangle in the coordinate plane.
Consider a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3).
The area of the triangle is given by 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.

Example: Area of a Triangle

Consider a triangle with vertices A(1, 3), B(4, -2), and C(-2, 5).

Area = 0.5 * |1(-2 - 5) + 4(5 - 3) + (-2)(3 - (-2))|
Area = 0.5 * |-7 + 8 - 10|
Area = 0.5 * |-9|
Area = 4.5

Determinants and Parallel Lines

Determinants can also be used to check if two lines are parallel.
Given two lines Ax + By = C and Dx + Ey = F, the lines are parallel if and only if the determinant A * E - B * D = 0.

Example: Checking Parallel Lines

Check if the lines 2x + 3y = 5 and 4x + 6y = 10 are parallel.

Let A = 2, B = 3, D = 4, and E = 6.
The determinant A * E - B * D = 2 * 6 - 3 * 4
Determinant = 12 - 12
Determinant = 0 Since the determinant is zero, the lines are parallel.

Determinants and Cramer’s Rule

Determinants can also be used to solve systems of linear equations using Cramer’s Rule.
Given a system of equations in the form Ax + By = C and Dx + Ey = F, the solution for x and y can be found using the determinants of the coefficients.
x = (C * E - B * F) / (A * E - B * D)
y = (A * F - C * D) / (A * E - B * D)

Example: Solving Equations using Cramer’s Rule

Solve the system of equations:

2x + 3y = 7

4x + 2y = 10

Let A = 2, B = 3, C = 7, D = 4, and E = 2, F = 10.
Calculate the determinants:
- Main determinant = (2 * 2) - (3 * 4) = -2
- Determinant for x = (7 * 2) - (3 * 10) = -8
- Determinant for y = (2 * 10) - (7 * 4) = 6
x = -8 / -2 = 4
y = 6 / -2 = -3 The solution for x and y is x = 4, y = -3.

Summary

Determinants can be used to find the area of triangles.
Determinants can be used to check if two lines are parallel.
Determinants can be used to solve systems of linear equations using Cramer’s Rule.
Practice calculating determinants and applying them to different problems.

Determinants - Determinants and Area of triangles

Determinants can be used to find the area of triangles.
The formula for the area of a triangle using determinants is 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.
The vertices of the triangle are represented as (x1, y1), (x2, y2), and (x3, y3).
The absolute value of the determinant represents the area of the triangle.
If the determinant is negative, it means the vertices are in clockwise order.

Example: Finding the Area of a Triangle

Consider a triangle with the vertices A(2, 3), B(-1, 4), and C(0, -2).

Using the formula, the area of the triangle can be calculated as:
- Area = 0.5 * |2(4 - (-2)) + (-1)(-2 - 3) + 0(3 - 4)|
- Area = 0.5 * |2(6) + (-1)(-5) + 0(-1)|
- Area = 0.5 * |12 + 5 + 0|
- Area = 0.5 * |17|
- Area = 8.5 So, the area of the triangle ABC is 8.5 square units.

Determinants - Determinants and Volumes of Parallelepiped

Determinants can also be used to find the volume of parallelepiped.
A parallelepiped is a three-dimensional figure with six parallelogram faces.
The volume of a parallelepiped with sides represented by vectors a, b, and c is given by the absolute value of the determinant of the matrix [a b c].

Example: Finding the Volume of a Parallelepiped

Consider a parallelepiped with sides represented by vectors a = [2, 0, -1], b = [3, 1, 4], and c = [-1, 2, 1]. To find the volume of the parallelepiped, we calculate the determinant of the matrix [a b c]:

Volume = |2 0 -1| |3 1 4| |-1 2 1|
Volume = 2(1 * 1 - 2 * 4) - 0(3 * 1 - (-1) * 4) - (-1)(3 * 2 - (-1) * 1) Simplifying the expression:
Volume = 2(-7) - 0(7) - (-1)(5)
Volume = -14 - 0 - (-5)
Volume = -9 The volume of the parallelepiped is -9 cubic units.

Determinants - Properties of Determinants

Determinants have several important properties:
1. The determinant of the identity matrix is 1.
2. If two rows or columns of a matrix are interchanged, the sign of the determinant changes.
3. If two rows or columns of a matrix are multiplied by a scalar, the determinant is multiplied by the same scalar.
4. If two rows or columns of a matrix are equal, the determinant is zero.
5. If one row or column of a matrix is a multiple of another row or column, the determinant is zero.

Example: Applying Properties of Determinants

Consider the matrix A = [2 3; 1 4]. Let’s apply the properties of determinants to find the determinant of A:

The determinant of the identity matrix is 1.
- Determinant(A) = 2 * 4 - 3 * 1 = 8 - 3 = 5

If two rows or columns of a matrix are interchanged, the sign of the determinant changes.
- If we interchange the rows of A, the determinant changes sign:
  - Determinant(interchanged A) = 4 * 3 - 1 * 2 = 12 - 2 = 10

If two rows or columns of a matrix are multiplied by a scalar, the determinant is multiplied by the same scalar.
- If we multiply the first row of A by 2, the determinant is also multiplied by 2:
  - Determinant(2A) = (2 * 2)(4 * 1) - (2 * 3)(1 * 1) = 16 - 6 = 10

If two rows or columns of a matrix are equal, the determinant is zero.
- If we make the first row of A equal to the second row, the determinant becomes zero:
  - Determinant(equal rows A) = 0

If one row or column of a matrix is a multiple of another row or column, the determinant is zero.
- If the second row of A is -2 times the first row, the determinant becomes zero:
  - Determinant(multiple rows A) = 0

Determinants - Finding Inverse of a Matrix

Determinants can be used to find the inverse of a matrix.
A matrix A is invertible (or non-singular) if its determinant is non-zero.
If A is invertible, the inverse of A (denoted as A^-1) can be found using the formula A^-1 = (1 / Determinant(A)) * Adjoint(A), where Adjoint(A) is the transpose of the cofactor matrix of A.

Example: Finding the Inverse of a Matrix

Consider the matrix A = [3 2; 1 4]. To find the inverse of A, we need to calculate the determinant of A:

Determinant(A) = 3 * 4 - 2 * 1 = 12 - 2 = 10 Since the determinant is non-zero, A is invertible. Next, we need to calculate the adjoint of A:
Adjoint(A) = [4 -2; -1 3] Finally, we can find the inverse of A using the formula:
A^-1 = (1 / Determinant(A)) * Adjoint(A)
A^-1 = (1 / 10) * [4 -2; -1 3]
A^-1 = [0.4 -0.2; -0.1 0.3] So, the inverse of matrix A is [0.4 -0.2; -0.1 0.3].

Determinants - Applications in Solving Systems of Equations

Determinants can be used to solve systems of linear equations.
Given a system of equations in the form Ax + By + Cz = D, Ex + Fy + Gz = H, and Ix + Jy + Kz = L, we can write the system as a matrix equation [A B C; E F G; I J K] * [x; y; z] = [D; H; L].
If the determinant of the coefficient matrix is non-zero, the system has a unique solution.
If the determinant is zero, the system may have infinitely many solutions or no solution at all.

Example: Solving a System of Equations using Determinants

Consider the system of equations:

2x + 3y - z = 7, x + 4y + 2z = 1,

3x - 2y - z = 3. To solve this system using determinants, we set up the coefficient matrix and the solution matrix: Coefficient matrix: [2 3 -1; 1 4 2; 3 -2 -1] Solution matrix: [7; 1; 3] Next, we calculate the determinant of the coefficient matrix: Determinant = 2(4 * -1 - 2 * -2) - 3(1 * -1 - 2 * 3) + (-1)(1 * -2 - 4 * 3) Determinant = 2(8 + 4) - 3(-1 - 6) - (-1)(-2 - 12) Determinant = 2(12) - 3(-7) + 1(10) Determinant = 24 + 21 + 10 Determinant = 55 Since the determinant is non-zero, the system has a unique solution. To find the solution, we calculate the determinants of the matrices formed by replacing each column with the solution matrix: Determinant x = [7 3 -1; 1 1 2; 3 -2 -1] Determinant y = [2 7 -1; 1 1 2; 3 3 -1] Determinant z = [2 3 7; 1 4 1; 3 -2 3] Simplifying the determinants and dividing by the main determinant: x = (Determinant x) / Determinant y = (Determinant y) / Determinant z = (Determinant z) / Determinant After solving these equations, we find the values of x, y, and z that satisfy the system of equations.

Determinants - Determinants and Area of triangles

Determinants can be used to find the area of triangles.
The formula for the area of a triangle using determinants is 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|.
The vertices of the triangle are represented as (x1, y1), (x2, y2), and (x3, y3).
The absolute value of the determinant represents the area of the triangle.
If the determinant is negative, it means the vertices are in clockwise order.

Example: Finding the Area of a Triangle

Consider a triangle with the vertices A(2, 3), B(-1, 4), and C(0, -2). To find the area of the triangle, we can use the formula:

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|
Area = 0.5 * |2(4 - (-2)) + (-1)(-2 - 3) + 0(3 - 4)|
Area = 0.5 * |2(6) + (-1)(-5) + 0(-1)|
Area = 0.5 * |12 + 5 + 0|
Area = 0.5 * |17|
Area = 8.5 So, the area of the triangle ABC is 8.5 square units.

Determinants - Determinants and Volumes of Parallelepiped

Determinants can also be used to find the volume of parallelepiped.
A parallelepiped is a three-dimensional figure with six parallelogram faces.
The volume of a parallelepiped with sides represented by vectors a, b, and c is given by the absolute value of the determinant of the matrix [a b c].

Example: Finding the Volume of a Parallelepiped

Volume = |2 0 -1| |3 1 4| |-1 2 1|
Volume = 2(1 * 1 - 2 * 4) - 0(3 * 1 - (-1) * 4) - (-1)(3 * 2 - (-1) * 1) Simplifying the expression:
Volume = 2(-7) - 0(7) - (-1)(5)
Volume = -14 - 0 - (-5)
Volume = -9 The volume of the parallelepiped is -9 cubic units.

Determinants - Properties of Determinants

Determinants have several important properties:
1. The determinant of the identity matrix is 1.
2. If two rows or columns of a matrix are interchanged, the sign of the determinant changes.
3. If two rows or columns of a matrix are multiplied by a scalar, the determinant is multiplied by the same scalar.
4. If two rows or columns of a matrix are equal, the determinant is zero.
5. If one row or column of a matrix is a multiple of another row or column, the determinant is zero.

Example: Applying Properties of Determinants

Consider the matrix A = [2 3; 1 4]. Let’s apply the properties of determinants