Determinants - Introduction to determinants

Determinants - Introduction to determinants

Slide 1

  • The determinant is a mathematical concept used in linear algebra.
  • It is a scalar value that is derived from a matrix.
  • It provides information about the matrix and its properties.

Slide 2

  • Determinants are denoted using vertical lines or brackets.
  • For example: If A is a matrix, then the determinant of A is represented as |A| or det(A).

Slide 3

  • The determinant of a matrix can only be computed for square matrices.
  • A square matrix has an equal number of rows and columns.

Slide 4

  • The order of a square matrix refers to the number of rows (or columns) it has.
  • For example: If a matrix has 3 rows and 3 columns, it is a 3x3 matrix.

Slide 5

  • To compute the determinant of a 2x2 matrix, use the formula:
    • |A| = ad - bc,
    • where A = [ a b ; c d ]

Slide 6

  • Example:
    • Compute the determinant of the matrix A = [ 2 4 ; 3 1 ]

Slide 7

  • Solution:
    • Using the formula |A| = ad - bc,
    • We have |A| = (2 * 1) - (4 * 3) = 2 - 12 = -10.

Slide 8

  • To compute the determinant of a 3x3 matrix, use the rule of expansion by minors.
  • The rule of expansion states that the determinant of a matrix can be computed by expanding along any row or any column.

Slide 9

  • Example:
    • Compute the determinant of the matrix B = [ 1 2 3 ; 4 5 6 ; 7 8 9 ]

Slide 10

  • Solution:
    • Expand along the first row,
    • |B| = (1 * |M1|) - (2 * |M2|) + (3 * |M3|),
    • where M1, M2, and M3 are the 2x2 matrices obtained by removing the first row and column, second row and column, and third row and column respectively.

Slide 11

  • Continuing from the previous slide,
  • We have M1 = [5 6; 8 9],
  • M2 = [4 6; 7 9], and
  • M3 = [4 5; 7 8].

Slide 12

  • Compute the determinants of M1, M2, and M3 using the formula |A| = ad - bc.

Slide 13

  • Solution:
    • |M1| = (5 * 9) - (6 * 8) = 45 - 48 = -3,
    • |M2| = (4 * 9) - (6 * 7) = 36 - 42 = -6,
    • |M3| = (4 * 8) - (5 * 7) = 32 - 35 = -3.

Slide 14

  • Substitute the determinants of M1, M2, and M3 into the equation,
  • |B| = (1 * (-3)) - (2 * (-6)) + (3 * (-3)).

Slide 15

  • Simplify the equation to get the final value of the determinant |B|.

Slide 16

  • Solution:
    • |B| = (-3) - (-12) + (-9) = -3 + 12 + (-9) = 0.

Slide 17

  • In general, for an n x n matrix,
  • The determinant can be computed using the rule of expansion by minors.

Slide 18

  • The rule of expansion by minors involves expanding the determinant by cofactors.

Slide 19

  • The cofactor of an element is the product of the element and the determinant of the matrix after removing the row and column containing the element.

Slide 20

  • Example:
    • Compute the determinant of the matrix C = [ 1 2 3 4 ; 5 6 7 8 ; 9 10 11 12 ; 13 14 15 16 ]

Slide 21

  • Determinants can be used to solve systems of linear equations.
  • If the determinant of a matrix is non-zero, the system has a unique solution.
  • If the determinant is zero, the system has either no solution or infinitely many solutions.

Slide 22

  • The value of the determinant provides information about the matrix.
  • If the determinant is positive, the matrix is said to be positive-definite.
  • If the determinant is negative, the matrix is said to be negative-definite.

Slide 23

  • If the determinant is zero, the matrix is said to be singular.
  • A singular matrix does not have an inverse.
  • Non-zero determinant implies the existence of an inverse.

Slide 24

  • Determinants can be used to find the area of a parallelogram or the volume of a parallelepiped.
  • The absolute value of the determinant gives the magnitude of the area/volume.

Slide 25

  • Determinants can also be used to find the eigenvalues and eigenvectors of a matrix.
  • Eigenvalues are the scalar values that satisfy the equation Av = λv, where A is the matrix, λ is the eigenvalue, and v is the eigenvector.

Slide 26

  • Determinants follow certain properties:
    • Swapping the rows (or columns) of a matrix changes the sign of the determinant.
    • Multiplying a row (or column) by a scalar multiplies the determinant by the same scalar.
    • Adding a multiple of one row (or column) to another row (or column) does not change the determinant.

Slide 27

  • The determinant of a matrix can be calculated using various methods:
    • Expansion by minors.
    • Row reduction.
    • Cofactor expansion.
    • Using properties of determinants.

Slide 28

  • Determinants have many applications in various fields of science and engineering.
  • They are used in solving systems of linear equations, analyzing linear transformations, and computer graphics.

Slide 29

  • Practice problem:
    • Compute the determinant of the matrix D = [ 3 1 2 ; 2 4 1 ; 5 2 3 ]

Slide 30

  • Summary:
    • Determinants provide information about matrices and their properties.
    • They can be computed for square matrices.
    • Determinants are used in solving systems of linear equations, finding areas/volumes, and calculating eigenvalues/eigenvectors.
    • They follow certain properties and can be calculated using various methods.