Determinants - What are Determinants

  • Determinants are specific values that can be calculated from the elements of a square matrix.
  • Determinants play a crucial role in many areas of Mathematics, including solving systems of linear equations, finding inverses of matrices, and determining the properties of vectors and transformations.
  • In this lecture, we will explore the concept of determinants and understand their significance.

Determinant of a 2x2 Matrix

  • A 2x2 matrix has four elements arranged in two rows and two columns.
  • The determinant of a 2x2 matrix A, denoted as |A| or det(A), can be calculated as follows:
    • |A| = (ad) - (bc), where a, b, c, and d are the elements of the matrix. Example: Given matrix A = [[2, 3], [4, 5]] |A| = (2*5) - (3*4) = 10 - 12 = -2

Determinant of a 3x3 Matrix

  • A 3x3 matrix has nine elements arranged in three rows and three columns.
  • The determinant of a 3x3 matrix A, denoted as |A| or det(A), can be calculated using the Laplace Expansion or the Sarrus Rule. Laplace Expansion method: |A| = a(ei - fh) - b(di - fg) + c(dh - eg), where a, b, c, d, e, f, g, h, and i are the elements of the matrix. Sarrus Rule method: |A| = (a*e*i + b*f*g + c*d*h) - (c*e*g + b*d*i + a*f*h) Example: Given matrix A = [[2, 1, 5], [3, 4, 2], [1, 6, 3]] |A| = (2*(4*3 - 2*6) - 1*(3*3 - 1*6) + 5*(3*6 - 4*1))

Properties of Determinants

  • Determinants exhibit several important properties that are useful in matrix computations.
  • Property 1: If any two rows (or columns) of a matrix A are interchanged, the determinant changes its sign.
  • Property 2: If any row (or column) of a matrix A is multiplied by a scalar k, the determinant is multiplied by the same scalar.
  • Property 3: If any two rows (or columns) of a matrix A are identical, the determinant is zero.
  • Property 4: If a matrix A has a row (or column) consisting of all zeros, the determinant is zero.
  • Property 5: The determinant of an identity matrix is 1. Example: Given matrix A = [[3, 1, 4], [2, 0, 5], [1, 2, 3]] |A| = 0, because the second row consists of all zeros.

Cofactors and Cofactor Expansion

  • The cofactor of an element aij of a matrix A, denoted as Cij, is calculated as follows:
    • Cij = (-1)^(i+j) * |Mij|, where Mij is the determinant of the matrix obtained by deleting the ith row and jth column of A.
  • The determinant of a matrix A can be calculated using the cofactor expansion along any row or column.
  • The cofactor expansion along a row can be written as follows:
    • |A| = a1C1 + a2C2 + a3C3 + … + anCn, where a1, a2, …, an are the elements of the selected row. Example: Given matrix A = [[2, 1, 5], [3, 4, 2], [1, 6, 3]] |A| = 2*C11 - 1*C12 + 5*C13, where C11, C12, and C13 are the cofactors of the first row.

Determinant of an Upper Triangular Matrix

  • An upper triangular matrix is a square matrix in which all elements below the main diagonal are zeros.
  • The determinant of an upper triangular matrix is equal to the product of its diagonal elements. Example: Given matrix A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]] |A| = 1*4*6 = 24

Determinant of a Lower Triangular Matrix

  • A lower triangular matrix is a square matrix in which all elements above the main diagonal are zeros.
  • The determinant of a lower triangular matrix is equal to the product of its diagonal elements. Example: Given matrix A = [[1, 0, 0], [2, 3, 0], [4, 5, 6]] |A| = 1*3*6 = 18

Determinant of a Diagonal Matrix

  • A diagonal matrix is a square matrix in which all elements outside the main diagonal are zeros.
  • The determinant of a diagonal matrix is equal to the product of its diagonal elements. Example: Given matrix A = [[2, 0, 0], [0, -1, 0], [0, 0, 5]] |A| = 2*(-1)*5 = -10

Determinant of a Singular Matrix

  • A singular matrix, also known as a non-invertible matrix, is a square matrix with a determinant of zero.
  • A singular matrix does not have an inverse and cannot be used to solve systems of linear equations. Example: Given matrix A = [[1, 2], [2, 4]] |A| = 0, because the second row is a scalar multiple of the first row.

Keep watching for more information on determinants and their applications.

Determinants - What are Determinants

  • Determinants are specific values that can be calculated from the elements of a square matrix.
  • Determinants play a crucial role in many areas of Mathematics, including solving systems of linear equations, finding inverses of matrices, and determining the properties of vectors and transformations.
  • In this lecture, we will explore the concept of determinants and understand their significance.

Determinant of a 2x2 Matrix

  • A 2x2 matrix has four elements arranged in two rows and two columns.
  • The determinant of a 2x2 matrix A, denoted as |A| or det(A), can be calculated as follows:
    • |A| = (ad) - (bc), where a, b, c, and d are the elements of the matrix.
  • Example:
    • Given matrix A = [[2, 3], [4, 5]]
    • |A| = (25) - (34) = 10 - 12 = -2

Determinant of a 3x3 Matrix

  • A 3x3 matrix has nine elements arranged in three rows and three columns.
  • The determinant of a 3x3 matrix A can be calculated using either the Laplace Expansion or the Sarrus Rule.
  • Laplace Expansion method:
    • |A| = a(ei - fh) - b(di - fg) + c(dh - eg), where a, b, c, d, e, f, g, h, and i are the elements of the matrix.
  • Sarrus Rule method:
    • |A| = (aei + bfg + cdh) - (ceg + bdi + afh)
  • Example:
    • Given matrix A = [[2, 1, 5], [3, 4, 2], [1, 6, 3]]
    • |A| = (2*(43 - 26) - 1*(33 - 16) + 5*(36 - 41))

Properties of Determinants

  • Determinants exhibit several important properties that are useful in matrix computations.
  • Property 1: If any two rows (or columns) of a matrix A are interchanged, the determinant changes its sign.
  • Property 2: If any row (or column) of a matrix A is multiplied by a scalar k, the determinant is multiplied by the same scalar.
  • Property 3: If any two rows (or columns) of a matrix A are identical, the determinant is zero.
  • Property 4: If a matrix A has a row (or column) consisting of all zeros, the determinant is zero.
  • Property 5: The determinant of an identity matrix is 1.
  • Example:
    • Given matrix A = [[3, 1, 4], [2, 0, 5], [1, 2, 3]]
    • |A| = 0, because the second row consists of all zeros.

Cofactors and Cofactor Expansion

  • The cofactor of an element aij of a matrix A, denoted as Cij, is calculated as follows:
    • Cij = (-1)^(i+j) * |Mij|, where Mij is the determinant of the matrix obtained by deleting the ith row and jth column of A.
  • The determinant of a matrix A can be calculated using the cofactor expansion along any row or column.
  • The cofactor expansion along a row can be written as follows:
    • |A| = a1C1 + a2C2 + a3C3 + … + anCn, where a1, a2, …, an are the elements of the selected row.
  • Example:
    • Given matrix A = [[2, 1, 5], [3, 4, 2], [1, 6, 3]]
    • |A| = 2C11 - 1C12 + 5*C13, where C11, C12, and C13 are the cofactors of the first row.

Determinant of an Upper Triangular Matrix

  • An upper triangular matrix is a square matrix in which all elements below the main diagonal are zeros.
  • The determinant of an upper triangular matrix is equal to the product of its diagonal elements.
  • Example:
    • Given matrix A = [[1, 2, 3], [0, 4, 5], [0, 0, 6]]
    • |A| = 146 = 24

Determinant of a Lower Triangular Matrix

  • A lower triangular matrix is a square matrix in which all elements above the main diagonal are zeros.
  • The determinant of a lower triangular matrix is equal to the product of its diagonal elements.
  • Example:
    • Given matrix A = [[1, 0, 0], [2, 3, 0], [4, 5, 6]]
    • |A| = 136 = 18

Determinant of a Diagonal Matrix

  • A diagonal matrix is a square matrix in which all elements outside the main diagonal are zeros.
  • The determinant of a diagonal matrix is equal to the product of its diagonal elements.
  • Example:
    • Given matrix A = [[2, 0, 0], [0, -1, 0], [0, 0, 5]]
    • |A| = 2*(-1)*5 = -10

Determinant of a Singular Matrix

  • A singular matrix, also known as a non-invertible matrix, is a square matrix with a determinant of zero.
  • A singular matrix does not have an inverse and cannot be used to solve systems of linear equations.
  • Example:
    • Given matrix A = [[1, 2], [2, 4]]
    • |A| = 0, because the second row is a scalar multiple of the first row. Keep watching for more information on determinants and their applications.

Determinants - Properties of Determinants (Contd.)

Property 6: Determinant of the Transpose of a Matrix

  • The determinant of the transpose of a matrix A is equal to the determinant of matrix A.
  • Symbolically, |A^T| = |A|.
  • Example:
    • Given matrix A = [[2, 3], [4, 5]]
    • |A^T| = |A| = (25) - (34) = 10 - 12 = -2

Property 7: Determinant of the Product of Matrices

  • The determinant of the product of two matrices A and B is equal to the product of their determinants.
  • Symbolically, |A * B| = |A| * |B|.
  • Example:
    • Given matrix A = [[2, 3], [4, 5]] and matrix B = [[1, 2], [3, 4]]
    • |A * B| = |A| * |B| = (-2) * (-2) = 4

Property 8: Determinant of the Inverse of a Matrix

  • The determinant of the inverse of a matrix A is equal to the reciprocal of the determinant of matrix A.
  • Symbolically, |A^(-1)| = 1 / |A|.
  • Example:
    • Given matrix A = [[2, 3], [4, 5]]
    • Calculate |A^(-1)| and compare it with 1 / |A|.

Property 9: Determinant of a Block Matrix

  • If a matrix A is divided into blocks, the determinant of matrix A can be calculated using the block determinants.
  • Example:
    • Given block matrix [[A, B], [C, D]]
    • Calculate the determinant of the block matrix using the determinants of submatrices A, B, C, and D.

Property 10: Determinant of the Sum of Matrices

  • The determinant of the sum of two matrices A and B cannot be determined solely from their individual determinants.
  • Symbolically, |A + B| ≠ |A| + |B|.
  • Example:
    • Given matrix A = [[2, 3], [4, 5]] and matrix B = [[1, -2], [3, 7]]
    • Calculate |A + B| and compare it with |A| + |B|.

Determinants - Applications of Determinants

Application 1: Solving Systems of Linear Equations

  • Determinants can be used to solve systems of linear equations.
  • The Cramer’s Rule states that if |A| ≠ 0, then the system of linear equations has a unique solution given by:
    • x = |A_x| / |A|, y = |A_y| / |A|, z = |A_z| / |A|, where |A_x|, |A_y|, and |A_z| are determinants obtained by replacing the coefficients of variable x, y, and z in the system.
  • Example:
    • Solve the following system of linear equations using determinants:
      • 2x + 3y + z = 9
      • x - 2y + 3z = 1
      • 3x + y - z = 7

Application 2: Checking Linear Independence

  • Determinants can be used to check the linear independence of vectors.
  • If the determinant of a matrix formed by the vectors is zero, the vectors are linearly dependent.
  • If the determinant is non-zero, the vectors are linearly independent.
  • Example:
    • Determine if the vectors [1, 2, 3], [4, 5, 6], and [7, 8, 9] are linearly independent.

Application 3: Finding Area of a Parallelogram

  • The area of a parallelogram formed by two vectors A and B can be calculated using the magnitude of their cross product.
  • Symbolically, Area = |A × B|.
  • Example:
    • Given vectors A = [2, 3] and B = [4, 5]
    • Calculate the area of the parallelogram formed by vectors A and B.

Application 4: Calculating Volume of a Parallelepiped

  • The volume of a parallelepiped formed by three vectors A, B, and C can be calculated using the scalar triple product.
  • Symbolically, Volume = |A ⋅ (B × C)|.
  • Example:
    • Given vectors A = [2, 3, 4], B = [1, -2, 3], and C = [3, 1, -2]
    • Calculate the volume of the parallelepiped formed by vectors A, B, and C.

Application 5: Finding Eigenvalues and Eigenvectors

  • Determinants can be used to find the eigenvalues and eigenvectors of a matrix.
  • The eigenvalues are the values λ such that det(A - λI) = 0, where A is the matrix and I is the identity matrix.
  • The eigenvectors are the non-zero vectors v satisfying Av = λv, where A is the matrix and λ is the corresponding eigenvalue.
  • Example:
    • Find the eigenvalues and eigenvectors of the matrix A = [[2, 1], [4, 3]].

Keep watching for more applications and interesting concepts related to determinants.