Determinants - Inverse of a Matrix

  • Introduction to determinants and inverse of a matrix
  • What is a determinant?
  • Calculating the determinant of a 2x2 matrix
  • Calculating the determinant of a 3x3 matrix
  • Properties of determinants
  • What is an inverse matrix?
  • How to find the inverse of a matrix?
  • Example: Finding the inverse of a 2x2 matrix
  • Example: Finding the inverse of a 3x3 matrix
  • Solving linear equations using inverse matrices

Introduction to Determinants and Inverse of a Matrix

  • Determinants and inverse of a matrix are important concepts in linear algebra.
  • They have significant applications in various fields such as physics, engineering, and computer science.
  • In this lecture, we will learn what determinants and inverse matrices are and how to calculate them.

What is a Determinant?

  • A determinant is a scalar value associated with a square matrix.
  • It represents certain properties and characteristics of the matrix.
  • Determinants are denoted by the symbol |A|, where A is the matrix.

Calculating the Determinant of a 2x2 Matrix

  • The determinant of a 2x2 matrix can be calculated using the formula: |A| = ad - bc where A = [[a, b], [c, d]]
  • Example: A = [[2, 4], [-1, 3]] |A| = (2 * 3) - (4 * -1) = 6 + 4 = 10

Calculating the Determinant of a 3x3 Matrix

  • The determinant of a 3x3 matrix can be calculated using the formula: |A| = a(ei - fh) - b(di - fg) + c(dh - eg) where A = [[a, b, c], [d, e, f], [g, h, i]]
  • Example: A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]] |A| = 1(59 - 68) - 2(49 - 67) + 3(48 - 57) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = 1*(-3) - 2*(-6) + 3*(-3) = -3 + 12 - 9 = 0

Properties of Determinants

  • Determinants have several properties that make them useful in various calculations.
  • Some of the properties include:
    1. The determinant of the identity matrix is always 1.
    2. Swapping rows or columns changes the sign of the determinant.
    3. Multiplying a row or column by a scalar multiplies the determinant by that scalar.
    4. If two rows or columns are equal, the determinant is zero.

What is an Inverse Matrix?

  • An inverse matrix is a matrix that, when multiplied with the original matrix, gives the identity matrix.
  • The inverse of a matrix A is denoted as A-1 or Ainv.

How to Find the Inverse of a Matrix?

  • The inverse of a square matrix A can be found using the following formula: A-1 = (1/|A|) * adj(A) where |A| is the determinant of A and adj(A) is the adjugate of A.
  • Note: Not all matrices have an inverse. Only non-singular matrices have inverses.

Example: Finding the Inverse of a 2x2 Matrix

  • Let’s find the inverse of matrix A: A = [[2, 1], [4, 3]]
  • Follow the steps:
    1. Calculate the determinant of A.
    2. Calculate the adjugate of A.
    3. Multiply the adjugate by (1/|A|).
    • Example (Continued on the next slide)

Example: Finding the Inverse of a 2x2 Matrix (Continued)

  • Example (Continued):
    1. |A| = (2 * 3) - (4 * 1) = 6 - 4 = 2
    2. Adjugate of A = [[3, -1], [-4, 2]]
    3. A-1 = (1/2) * [[3, -1], [-4, 2]] = [[3/2, -1/2], [-2, 1]]
  • Therefore, the inverse of matrix A is: A-1 = [[3/2, -1/2], [-2, 1]]

Example: Finding the Inverse of a 3x3 Matrix

  • Let’s find the inverse of matrix A: A = [[1, 0, 2], [0, 1, -3], [1, 1, -2]]
  • Follow the steps:
    1. Calculate the determinant of A.
    2. Calculate the adjugate of A.
    3. Multiply the adjugate by (1/|A|).
    • Example (Continued on the next slide)

Example: Finding the Inverse of a 3x3 Matrix (Continued)

  • Example (Continued):
    1. |A| = (1 * (1 * -2 - -3 * 1)) - (0 * (1 * -2 - -3 * 1)) + (2 * (1 * 1 - 0 * -3)) = (1 * -5) - (0 * -5) + (2 * 1) = -5 + 0 + 2 = -3
    2. Adjugate of A = [[-1, -1, 3], [2, 1, -2], [-3, -1, 1]]
    3. A-1 = (1/(-3)) * [[-1, -1, 3], [2, 1, -2], [-3, -1, 1]] = [[1/3, 1/3, -1], [-2/3, -1/3, 2/3], [1, 1/3, -1/3]]
  • Therefore, the inverse of matrix A is: A-1 = [[1/3, 1/3, -1], [-2/3, -1/3, 2/3], [1, 1/3, -1/3]]

Solving Linear Equations using Inverse Matrices

  • Inverse matrices can be used to solve systems of linear equations efficiently.
  • Given an equation AX = B, where A is the coefficient matrix, X is the unknown variable matrix, and B is the constant matrix, we can solve for X using the formula: X = A-1 * B
  • This provides us with the values of X that satisfy the system of equations.

These were the first 10 slides of the lecture on “Determinants - Inverse of a Matrix”. Sure! Here are slides 11 to 20 for the lecture on “Determinants - Inverse of a Matrix”:

  • Applications of determinants and inverse matrices
  • Determinants are used to solve systems of linear equations
  • Inverse matrices are used to find the solution of linear systems
  • Determinants are used in finding eigenvalues and eigenvectors
  • Inverse matrices are used in solving simultaneous equations
  • Solving a system of linear equations using determinants and inverse matrices
  • Given the equations:
    • 3x + 2y = 7
    • 4x + 5y = 8
  • Representing the equations in matrix form: AX = B
  • A = [[3, 2], [4, 5]] is the coefficient matrix
  • X = [[x], [y]] is the variable matrix
  • B = [[7], [8]] is the constant matrix
  • Calculate the determinant of matrix A
  • |A| = (3 * 5) - (2 * 4) = 15 - 8 = 7
  • Calculate the inverse of matrix A
  • A-1 = (1/|A|) * [[5, -2], [-4, 3]]
  • A-1 = (1/7) * [[5, -2], [-4, 3]] = [[5/7, -2/7], [-4/7, 3/7]]
  • Using the formula X = A-1 * B, substitute the values of A-1 and B
  • X = [[5/7, -2/7], [-4/7, 3/7]] * [[7], [8]]
  • X = [[(5/7) * 7 + (-2/7) * 8], [(-4/7) * 7 + (3/7) * 8]]
  • X = [[5 - 16/7], [-4 + 24/7]]
  • X = [[35/7 - 16/7], [-28/7 + 24/7]]
  • X = [[19/7], [-4/7]]
  • Therefore, the solution to the system of equations is:
    • x = 19/7
    • y = -4/7
  • We have found the values of x and y that satisfy both equations
  • Non-invertible matrices and their significance
  • Non-invertible matrices are also known as singular matrices
  • These matrices have a determinant equal to zero
  • Singular matrices do not have an inverse
  • They represent situations where a unique solution does not exist or the equations are dependent
  • Example of a non-invertible matrix
  • A = [[2, 4], [1, 2]]
  • Calculate the determinant of A
  • |A| = (2 * 2) - (4 * 1) = 0
  • Since the determinant is zero, matrix A is non-invertible
  • Determinants and cross product in vector algebra
  • The magnitude of the cross product of two vectors can be computed using determinants
  • Given vectors v = [a, b, c] and u = [d, e, f]
  • The cross product v x u can be represented as the determinant:
    • v x u = |i j k | |a b c | |d e f|
  • Calculate the determinant to find the magnitude of the cross product
  • Example of computing the cross product
  • v = [1, 2, 3] and u = [4, 5, 6]
  • v x u = |i j k | |1 2 3 | |4 5 6|
  • v x u = (2 * 6 - 3 * 5)i - (1 * 6 - 3 * 4)j + (1 * 5 - 2 * 4)k
  • v x u = (12 - 15)i - (6 - 12)j + (5 - 8)k
  • v x u = -3i - (-6)j - 3k = -3i + 6j - 3k
  • Summary of the lecture:
    • Determinants and inverse of matrices are important concepts in linear algebra
    • Determinants represent properties and characteristics of a matrix
    • Inverse matrices are used to solve systems of linear equations efficiently
    • Non-invertible matrices have determinant equal to zero and do not have an inverse
    • Determinants are also used in vector algebra, such as computing cross products This completes slides 11 to 20 for the lecture on “Determinants - Inverse of a Matrix”.
  • Cramer’s Rule for solving systems of linear equations
    • Cramer’s Rule states that if a system of linear equations can be written in the form AX = B, where A is the coefficient matrix, X is the unknown variable matrix, and B is the constant matrix, then the solution for X can be found using determinants.
  • Cramer’s Rule (cont.)
    • To solve for X using Cramer’s Rule, we substitute each column of matrix B into the corresponding column of matrix A, and calculate the determinant of each resulting matrix.
    • The value of X for each variable can be found by dividing the determinant of the matrix obtained for that variable by the determinant of the coefficient matrix.
  • Example: Solving a system of linear equations using Cramer’s Rule
    • Given the equations:
      • 2x + 3y = 7
      • 4x + 5y = 8
    • Representing the equations in matrix form: AX = B
    • A = [[2, 3], [4, 5]]
    • X = [[x], [y]]
    • B = [[7], [8]]
  • Calculate the determinant of the coefficient matrix A
    • |A| = (2 * 5) - (3 * 4) = 10 - 12 = -2
  • Calculate the determinants of the matrices obtained by substituting the columns of B into A
    • |Ax| = (7 * 5) - (3 * 8) = 35 - 24 = 11
    • |Ay| = (2 * 8) - (7 * 4) = 16 - 28 = -12
  • Calculate the values of x and y using Cramer’s Rule
    • x = |Ax| / |A| = 11 / -2 = -11/2
    • y = |Ay| / |A| = -12 / -2 = 6
  • Therefore, the solution to the system of equations is:
    • x = -11/2
    • y = 6
  • Cramer’s Rule provides a method to solve systems of linear equations using determinants, which can be especially useful for larger systems.
  • Matrix transformations and determinants
    • Matrices can represent transformations in various mathematical fields.
    • The determinant of a matrix transformation represents how the transformation affects the area or volume of a given shape.
    • If the determinant is zero, the transformation collapses the shape to a lower-dimensional object (e.g., a line to a point).
  • Matrix transformations and determinants (cont.)
    • If the determinant is positive, the transformation preserves orientation and does not change the shape or volume of the object.
    • If the determinant is negative, the transformation reverses orientation and changes the shape or volume of the object.
  • Example of a matrix transformation
    • Given the transformation matrix T:
      • T = [[-1, 0], [0, 2]]
    • Let’s apply this transformation to a square with vertices (0, 0), (1, 0), (1, 1), and (0, 1).
  • Example of a matrix transformation (cont.)
    • The transformed square has vertices (-1, 0), (-1, 2), (0, 2), and (0, 0).
    • The original square has an area of 1, while the transformed square has an area of 2.
    • The determinant of matrix T is 2, indicating that the transformation doubles the area of the shape.
  • Summary of the lecture:
    • Cramer’s Rule provides a method for solving systems of linear equations using determinants.
    • Determinants can be used to analyze the effects of matrix transformations on shapes.
    • Positive determinants preserve orientation and do not change shape or volume.
    • Negative determinants reverse orientation and change shape or volume. This completes slides 21 to 30 for the lecture on “Determinants - Inverse of a Matrix”.