Matrix And Determinant - Determinant Of Matrix

Matrix and Determinant - Determinant of Matrix

In linear algebra, a matrix is a rectangular array of numbers or symbols arranged in rows and columns.
The determinant of a square matrix is a scalar value that is computed from the elements of the matrix.
The determinant of a matrix is denoted as |A| or det(A), where A is the matrix.

Properties of Determinants

If A is a square matrix of order n, then the determinant of A is denoted as |A| or det(A).
The determinant of a 1x1 matrix is equal to the element of the matrix.
The determinant of a 2x2 matrix [ a b ; c d ] is equal to ad - bc.
The determinant of a matrix remains the same if its rows and columns are interchanged.
If two rows (or columns) of a matrix are identical, then its determinant is zero.
If a multiple of one row (or column) of a matrix is added to another row (or column), the determinant does not change.

Determinant of 3x3 Matrix

A 3x3 matrix can be represented as: [a b c] [d e f] [g h i] To calculate the determinant of a 3x3 matrix, use the formula: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Example

Let’s calculate the determinant of the matrix A: [2 1 5] [3 0 -2] [1 4 3] Using the determinant formula, we have: det(A) = 2(0*3 - (-2)*4) - 1(3*3 - (-2)*1) + 5(3*4 - 0*1) = 2(0 + 8) - 1(9 + 2) + 5(12) = 2(8) - 1(11) + 60 = 16 - 11 + 60 = 65 So, the determinant of matrix A is 65.

Determinants: Properties

The determinant of a matrix remains the same if we multiply all the elements of a row or column by a constant.
If we multiply all the elements of a row or column by a constant, the determinant gets multiplied by the same constant.
The determinant of the sum or difference of two matrices is equal to the sum or difference of their determinants.
The determinant of the product of two matrices is equal to the product of their determinants.
The determinant of the transpose of a matrix is equal to the determinant of the original matrix.

Determinants: Cofactor Expansion

The cofactor expansion method is used to calculate the determinant of a matrix.
In the cofactor expansion method, we expand the determinant along a row or column.
Let’s consider a 3x3 matrix A: [a b c; d e f; g h i]
The determinant of matrix A can be expanded along the first row as follows:
- |A| = a|A1| - b|A2| + c|A3|
- Where A1, A2, and A3 are 2x2 matrices obtained by removing the first row and the corresponding column from matrix A.

Determinants: Cofactor Expansion (Example 1)

Let’s calculate the determinant of the matrix A:
- A = [2 1 5; 3 0 -2; 1 4 3]
Expanding along the first row, we have:
- |A| = 2|A1| - 1|A2| + 5|A3|
- Where A1, A2, and A3 are obtained as follows:
  - A1 = [0 -2; 4 3], A2 = [3 -2; 1 3], A3 = [3 0; 1 4]
We can calculate the determinants of A1, A2, and A3 using the formula:
- |A1| = (03) - (-24) = 8, |A2| = (33) - (-21) = 11, |A3| = (34) - (01) = 12
Substituting the values, we get:
- |A| = 2(8) - 1(11) + 5(12) = 65

Determinants: Cofactor Expansion (Example 2)

Let’s calculate the determinant of the matrix B:
- B = [1 2 3; 0 4 5; -1 0 6]
Expanding along the second column, we have:
- |B| = 2|B1| + 4|B2| + 0|B3|
- Where B1, B2, and B3 are obtained as follows:
  - B1 = [0 5; -1 6], B2 = [1 3; -1 6], B3 = [1 3; 0 4]
We can calculate the determinants of B1, B2, and B3:
- |B1| = (06) - (5(-1)) = 5, |B2| = (16) - (3(-1)) = 9, |B3| = (14) - (30) = 4
Substituting the values, we get:
- |B| = 2(5) + 4(9) + 0(4) = 46

Determinants: Cramer’s Rule

Cramer’s Rule is a method to solve a system of linear equations using determinants.
Let’s consider a system of linear equations:
- a1x + b1y + c1z = d1
- a2x + b2y + c2z = d2
- a3x + b3y + c3z = d3
The solution for x, y, and z can be found using the following formulas:
- x = |Dx| / |A|
- y = |Dy| / |A|
- z = |Dz| / |A|
- Where Dx, Dy, and Dz are obtained by replacing the coefficients of x, y, and z respectively with d1, d2, and d3 in the original matrix A.

Determinants: Cramer’s Rule (Example)

Let’s solve the system of equations:
- 2x + 3y - z = 4
- x - 2y + 3z = -6
- 3x + 2y + 4z = 8
The coefficients can be represented in matrix form as:
- A = [2 3 -1; 1 -2 3; 3 2 4]
The constant terms can be represented in matrix form as:
- Dx = [4 3 -1; -6 -2 3; 8 2 4]
We can calculate the determinant of A and Dx using cofactor expansion:
- |A| = (2*(-24) + 3(34) - (-1)(3*2)) = 42
- |Dx| = (4*(-24) + 3(34) - (-1)(8*2)) = 38
Using Cramer’s Rule, we can find the solution:
- x = |Dx| / |A| = 38 / 42 = 19 / 21
- y = |Dy| / |A| = …
- z = |Dz| / |A| = …

Determinants: Inverse of a Matrix

The inverse of a square matrix A is denoted as A^-1.
A matrix A is said to be invertible if it exists an inverse A^-1 such that A * A^-1 = I, where I is the identity matrix.
To find the inverse of a matrix A, we use the following formula:
- A^-1 = (1 / |A|) * adj(A)
- Where |A| is the determinant of matrix A, and adj(A) is the adjugate of matrix A.
The inverse of a matrix can be used to solve systems of linear equations and perform various operations on matrices.

Determinants: Inverse of a Matrix (Example)

Let’s find the inverse of the matrix C:
- C = [2 1 -1; 3 4 1; 1 -2 3]
Firstly, we need to calculate the determinant of C:
- |C| = (2*(43) + 1(13) + (-1)(-2*1)) = 28
Next, we need to calculate the adjugate of C:
- C^T = [2 3 1; 1 4 -2; -1 1 3]
Finally, we can find the inverse of C using the formula:
- C^-1 = (1 / |C|) * C^T = (1 / 28) * [2 3 1; 1 4 -2; -1 1 3]

Determinants: Singular and Nonsingular Matrices

A square matrix A is called singular if its determinant is zero (|A| = 0).
A nonsingular or invertible matrix is a square matrix that is not singular.
A singular matrix does not have an inverse since the determinant is zero.
A nonsingular matrix has an inverse and can be used for various operations.
The existence of a unique solution in a system of linear equations is guaranteed if the coefficient matrix is nonsingular.

Determinants: Applications

Determinants have various applications in mathematics and other fields.
Determinants are used to find the inverse of a matrix.
Determinants are used in solving systems of linear equations through Cramer’s Rule.
Determinants are used to calculate the area of a parallelogram or triangle.
Determinants are used in finding the eigenvalues and eigenvectors of a matrix.
Determinants are used in solving differential equations and analyzing systems of differential equations.

Determinants: Area of Parallelogram

The determinant of a 2x2 matrix can be used to calculate the area of a parallelogram.
Let’s consider a parallelogram defined by vectors A and B.
The area of the parallelogram can be calculated as |A x B|, where x represents the cross product of A and B.

Determinants: Area of Parallelogram (Example)

Let’s calculate the area of a parallelogram defined by vectors A = [2, 3] and B = [4, -1].
The cross product of A and B can be calculated as:
- A x B = |i j | |2 3 | |4 -1 | = (2 * (-1)) - (3 * 4) = -2 - 12 = -14
The absolute value of the cross product represents the area of the parallelogram:
- Area = |-14| = 14

Determinants: Area of Triangle

The determinant of a 2x2 matrix can also be used to calculate the area of a triangle.
Let’s consider a triangle defined by vectors A and B.
The area of the triangle can be calculated as |(1/2) * (A x B)|, where x represents the cross product of A and B.

Determinants: Area of Triangle (Example)

Let’s calculate the area of a triangle defined by vectors A = [2, 3] and B = [4, -1].
The cross product of A and B can be calculated as -14 (as we calculated earlier).
The area of the triangle can be calculated as:
- Area = (1/2) * |-14| = (1/2) * 14 = 7

Determinants: Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are important concepts in linear algebra.
Let’s consider a square matrix A.
An eigenvector of A is a non-zero vector v such that Av = λv, where λ is a scalar called the eigenvalue.
In other words, multiplying matrix A with an eigenvector gives a scalar multiple of the same eigenvector.

Determinants: Eigenvalues and Eigenvectors (Example)

Let’s find the eigenvalues and corresponding eigenvectors of the matrix D:
- D = [3 2; 1 4]
To find the eigenvalues, we solve the equation |D - λI| = 0, where I is the identity matrix.
Expanding the determinant, we get:
- |D - λI| = |3-λ 2 ; 1 4-λ| = (3-λ)(4-λ) - (2)(1) = λ² - 7λ + 10
Solving the quadratic equation, we find the eigenvalues λ₁ = 5 and λ₂ = 2.
To find the eigenvectors, we substitute each eigenvalue into the equation Av = λv and solve for v.

Determinants: Differential Equations

Determinants are used in solving systems of differential equations.
Let’s consider a system of differential equations:
- dx/dt = 4x + 3y
- dy/dt = 2x + 5y
We can represent the system in matrix form as:
- [dx/dt; dy/dt] = [4 3; 2 5] [x; y]
Taking the determinant of the coefficient matrix, we get:
- |A| = (4 * 5) - (3 * 2) = 14
If the determinant is nonzero, a unique solution exists for the system of differential equations.

Determinants: Differential Equations (Example)

Let’s solve the system of differential equations using determinants:
- dx/dt = 2x - y
- dy/dt = x + 3y
The coefficient matrix is A = [2 -1; 1 3].
The determinant of A is |A| = (2 * 3) - (-1 * 1) = 7.
Since |A| ≠ 0, a unique solution exists for the system of differential equations.

Determinants: Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations.
The method involves using elementary row operations to transform the coefficient matrix into an upper triangular matrix.
Determinants can be used at each step of Gaussian elimination to check if the system has a unique solution.

Determinants: Gaussian Elimination (Example)

Let’s solve the system of equations using Gaussian elimination:
- 2x + 3y - z = 4
- x - 2y + 3z = -6
- 3x + 2y + 4z = 8
We start by writing the augmented matrix [A|B]:
- [2 3 -1 | 4]
- [1 -2 3 | -6]
- [3 2 4 | 8]
Applying elementary row operations, we can transform the matrix into an upper triangular form:
- [2 3 -1 | 4]
- [0 -7 5 | -14]
- [0 0 27 | 54]
At each step, we can calculate the determinants of the coefficient matrices to check if a unique solution exists.