Slide 1 - Determinants - Definition of Determinants, Minor, and Cofactor

  • A determinant is a scalar value that can be calculated for a square matrix.
  • Determinants play a crucial role in solving systems of linear equations, finding the inverse of a matrix, and many other mathematical applications.
  • The determinant of a matrix is denoted by |A| or det(A). Example: Consider the matrix A = [1 2; 3 4], the determinant |A| or det(A) can be calculated as follows: |A| = 1 * 4 - 2 * 3 = 4 - 6 = -2
  • The minor of an element aᵢⱼ in a square matrix A is denoted as Mᵢⱼ.
  • The minor of aᵢⱼ is obtained by deleting the ith row and jth column of the matrix A. Example: Consider the matrix A = [1 2; 3 4], the minor M₁₂ is obtained by deleting the first row and second column of the matrix A: M₁₂ = [3]
  • The cofactor of an element aᵢⱼ in a square matrix A is denoted as Cᵢⱼ.
  • The cofactor of aᵢⱼ is calculated as (-1)^(i+j) times the minor Mᵢⱼ. Example: Using the matrix A = [1 2; 3 4], the cofactor C₁₂ can be calculated as follows: C₁₂ = (-1)^(1+2) * M₁₂ = (-1)^3 * 3 = -3
  • The determinant of a matrix can be expressed in terms of the cofactors of the elements in a row or column. |A| = a₁₁C₁₁ + a₁₂C₁₂ + … + a₁ₙC₁ₙ (when expanding along the first row) |A| = aᵢ₁Cᵢ₁ + aᵢ₂Cᵢ₂ + … + aᵢₙCᵢₙ (when expanding along the ith row)
  • The value of the determinant remains the same when expanded along any row or column.

Slide 2 - Properties of Determinants

  • The determinant of a matrix can be calculated using various properties.
  • Let A, B, and C be square matrices of the same order, and k be a scalar.
  1. Property of Scalar Multiplication:
    • If A is a square matrix, then |kA| = k^n * |A|, where n is the order of A. Example: If k = 2 and A = [1 2; 3 4], then |2A| = 2^2 * |-2| = 4 * (-2) = -8.
  1. Property of Row/Column Addition or Subtraction:
    • If B is a matrix obtained from A by adding or subtracting any row/column of A multiplied by a scalar, then |B| = |A|. Example: If B is obtained from A = [1 2; 3 4] by adding 4 times the first column to the second column, then |B| = |A| = -2.
  1. Property of Interchanging Rows/Columns:
    • If B is obtained from A by interchanging any two rows/columns, then |B| = -|A|. Example: If A = [1 2; 3 4] and B is obtained by interchanging the first and second rows of A, then |B| = -|A| = 2.
  1. Property of Multiplicative Identity:
    • If A is a square matrix, then |I| = 1, where I is the identity matrix of the same order as A. Example: If A is a 3x3 identity matrix, then |A| = 1.
  1. Property of Multiplicative Property:
    • If A is a square matrix and |A| ≠ 0, then |A^(-1)| = 1/|A|, where A^(-1) is the inverse of A. Example: If A = [1 2; 3 4], then |A^(-1)| = 1/|-2| = -1/2. "
  1. Geometric Interpretation of Determinants
  • Determinants can also be interpreted geometrically in the context of vectors and transformations.
  • For a 2x2 matrix A, the determinant |A| represents the area scaling factor of the parallelogram formed by the column vectors of A.
  • If |A| > 0, then the transformation represented by A is an orientation-preserving transformation that does not flip or reverse the orientation of the vectors.
  • If |A| < 0, then the transformation represented by A is an orientation-reversing transformation that flips or reverses the orientation of the vectors. Example: Consider the 2x2 matrix A = [2 0; 0 -1]. The column vectors of A are [2, 0] and [0, -1]. The determinant |A| can be calculated as |A| = -2 * 0 - 0 * (-1) = 0. The area scaling factor of the parallelogram formed by the column vectors of A is 0, which means the vectors are linearly dependent and do not form a parallelogram.
  1. Solving Systems of Linear Equations using Determinants
  • Determinants can be used to solve systems of linear equations using Cramer’s Rule.
  • Cramer’s Rule states that if a system of linear equations can be expressed in the form AX = B, where A is a square matrix, X is a column vector of variables, and B is a column vector of constants, the solutions for X can be obtained by dividing the determinants of matrices formed by replacing the columns of A with B. Example: Consider the system of linear equations:

2x + 3y = 8

4x - 2y = 2 Using Cramer’s Rule, the determinant |A| = |2 3; 4 -2| = (2)(-2) - (3)(4) = -14. The determinant |A₁| is obtained by replacing the first column of A with the constants (8 and 2), |A₁| = |8 3; 2 -2| = (8)(-2) - (3)(2) = -22. The determinant |A₂| is obtained by replacing the second column of A with the constants, |A₂| = |2 8; 4 2| = (2)(2) - (8)(4) = -28. The solutions for x and y can be obtained by x = |A₁| / |A| and y = |A₂| / |A|.

  1. Inverse of a Matrix using Determinants
  • The inverse of a square matrix A can be obtained using determinants.
  • If |A| ≠ 0, then A^(-1) = (1/|A|) * adj(A), where A^(-1) is the inverse of A and adj(A) is the adjugate of A.
  • The adjugate of A is obtained by replacing each element of A with its cofactor, transposing the resulting matrix, and forming the adjugate matrix. Example: Consider the matrix A = [1 5; 2 3]. The determinant |A| can be calculated as |A| = (1)(3) - (5)(2) = -7. The adjugate of A, adj(A), is obtained by replacing each element with its cofactor, transposing the resulting matrix: adj(A) = [3 -5; -2 1] The inverse of A, A^(-1), can be obtained by multiplying adj(A) by (1/|A|): A^(-1) = (1/(-7)) * [3 -5; -2 1] = [-3/7 5/7; 2/7 -1/7]
  1. Properties of Inverse Matrices
  • Inverse matrices have several important properties.
  • If A is a square matrix and A^(-1) exists, then (A^(-1))^(-1) = A, meaning the inverse of an inverse is the original matrix.
  • If A and B are square matrices of the same order and A^(-1) and B^(-1) exist, then (AB)^(-1) = B^(-1)A^(-1), meaning the inverse of a product matrix is the product of inverses in reverse order.
  • If A is a square matrix and A^(-1) exists, then (kA)^(-1) = 1/k * A^(-1), where k is a scalar, meaning the inverse of a scalar multiple of a matrix is the reciprocal of the scalar multiplied by the inverse of the matrix.
  1. Area and Volume using Determinants
  • Determinants can be used to calculate the area of a triangle and the volume of a parallelepiped.
  • For a triangle with vertices A, B, and C in ℝ², the area of the triangle can be calculated as 1/2 times the absolute value of the determinant of the matrix formed by the coordinates of the vertices.
  • For a parallelepiped with vertices A, B, C, and D in ℝ³, the volume of the parallelepiped can be calculated as the absolute value of the determinant of the matrix formed by the coordinates of the vertices. Example 1: For a triangle with vertices A = (1, 2), B = (4, 5), and C = (3, 1), the area of the triangle is: Area = 1/2 * |1 4 3; 2 5 1; 1 1 1| = 1/2 * (1(5)(1) + 4(1)(1) + 3(2)(1) - 1(4)(1) - 3(5)(1) - 2(1)(1)) = 1/2 * (-3) = -3/2. Example 2: For a parallelepiped with vertices A = (1, 2, 3), B = (4, 5, 6), C = (7, 8, 9), and D = (10, 11, 12), the volume of the parallelepiped is: Volume = |1 4 7 10; 2 5 8 11; 3 6 9 12; 1 1 1 1| = 0, since the determinant is 0 and the parallelepiped is degenerate.
  1. Solving Homogeneous Systems of Linear Equations using Determinants
  • Homogeneous systems of linear equations have solutions that satisfy Ax = 0, where A is a matrix and x is a column vector of variables.
  • The solutions for x can be obtained by setting up the augmented matrix [A 0] and calculating the determinant of A.
  • If |A| ≠ 0, then the only solution is x = 0, meaning the system has only the trivial solution.
  • If |A| = 0, then the system has infinitely many non-trivial solutions, since the determinant of A is 0. Example: Consider the homogeneous system of linear equations:

2x - 4y = 0

3x + y = 0 Setting up the augmented matrix [A 0] gives:

2 -4 0

3 1 0 The determinant |A| = |2 -4; 3 1| = (2)(1) - (-4)(3) = 14. Since |A| ≠ 0, the only solution is x = y = 0.

  1. Solving Non-Homogeneous Systems of Linear Equations using Determinants and Cramer’s Rule
  • Non-homogeneous systems of linear equations have solutions that satisfy AX = B, where A is a matrix, X is a column vector of variables, and B is a column vector of constants.
  • The solutions for X can be obtained using Cramer’s Rule, which involves calculating the determinants of matrices formed by replacing the columns of A with B.
  • If |A| ≠ 0, then the system has a unique solution.
  • If |A| = 0, then the system either has no solution or infinitely many solutions, depending on the consistency of the system and the rank of the matrix A. Example: Consider the non-homogeneous system of linear equations:

2x - 4y = 8

3x + y = 2 Using Cramer’s Rule, the determinant |A| = |2 -4; 3 1| = (2)(1) - (-4)(3) = 14. The determinant |A₁| obtained by replacing the first column of A with the constants (8 and 2) is |A₁| = |8 -4; 2 1| = (8)(1) - (-4)(2) = 16. The determinant |A₂| obtained by replacing the second column of A with the constants is |A₂| = |2 8; 3 2| = (2)(2) - (8)(3) = -20. The solutions for x and y can be obtained by x = |A₁| / |A| and y = |A₂| / |A|.

  1. Solving Linear Dependence using Determinants
  • Determinants can be used to determine if a set of vectors is linearly dependent or linearly independent.
  • If the determinant of the matrix formed by the vectors is 0, then the vectors are linearly dependent.
  • If the determinant is non-zero, then the vectors are linearly independent. Example: Consider the vectors v₁ = [2; 3] and v₂ = [4; 6]. The matrix formed by the vectors is: [2 4] [3 6] The determinant |A| = |2 4; 3 6| = (2)(6) - (4)(3) = 12 - 12 = 0. Since |A| = 0, the vectors v₁ and v₂ are linearly dependent.
  1. Properties of Determinants for n x n Matrices
  • The properties of determinants discussed earlier for 2x2 and 3x3 matrices can be extended to n x n matrices.
  • The determinant of an n x n matrix A can be calculated through expansion along any row or column, using minors and cofactors.
  • The value of the determinant remains the same regardless of the row or column along which the expansion is performed. Example: For a 4x4 matrix A = [1 2 3 4; 5 6 7 8; 9 10 11 12; 13 14 15 16], we can calculate the determinant by expanding along the first row: |A| = 1 * C₁₁ - 2 * C₁₂ + 3 * C₁₃ - 4 * C₁₄ where C₁₁, C₁₂, C₁₃, and C₁₄ are the cofactors of the elements in the first row.
  1. Laplace’s Expansion Theorem
  • Laplace’s Expansion Theorem provides an alternate method to calculate the determinant of an n x n matrix A using minors and cofactors.
  • According to this theorem, the determinant of A can be obtained by expanding along any row or column and summing the products of the elements with their cofactors.
  • The expansion is performed recursively for each element in the row or column. Example: For a 3x3 matrix A = [1 2 3; 4 5 6; 7 8 9], we can calculate the determinant by expanding along the first row: |A| = 1 * C₁₁ - 2 * C₁₂ + 3 * C₁₃ where C₁₁, C₁₂, and C₁₃ are the cofactors of the elements in the first row. The cofactor C₁₁ is calculated as (-1)^(1+1) * M₁₁, where M₁₁ is the minor obtained from A by deleting the first row and first column. Similarly, the cofactors C₁₂ and C₁₃ are calculated for their respective minors. By substituting the values of the cofactors, we can calculate the determinant of A.
  1. Properties of Determinants cont.
  • If A and B are square matrices of the same order, then |AB| = |A| * |B|, meaning the determinant of a product matrix is the product of determinants.
  • If A is a square matrix and k is a scalar, then |kA| = k^n * |A|, where n is the order of A, meaning the determinant of a scalar multiple of a matrix is the scalar raised to the power of the order of the matrix multiplied by the determinant of the matrix.
  • If A is a square matrix and A^T is its transpose, then |A^T| = |A|, meaning the determinant of a matrix and its transpose are equal. Example 1: If A = [2 1; -3 4] and B = [5 -1; 2 3], then |AB| = |A| * |B| = |-11| * |17| = 187. Example 2: If A = [1 2; 3 4] and k = 3, then |kA| = k^n * |A| = 3^2 * |-2| = 18. Example 3: If A = [1 2; 3 4], then |A^T| = |A| = -2.
  1. Solving Systems of Linear Equations using Cramer’s Rule cont.
  • Cramer’s Rule can be used to solve systems of linear equations with three or more variables.
  • Given a system of linear equations in the form AX = B, where A is an n x n matrix, X is a column vector of variables, and B is a column vector of constants, the solutions for X can be obtained by dividing the determinants of matrices formed by replacing the columns of A with B. Example: Consider the system of linear equations:

2x + 3y - z = 9

3x - 2y + 4z = 1