Matrices - Elementary Row Operation and Row Echelon Matrix

  • Introduction to Matrices and its components
    • Definition: A rectangular array of numbers, symbols, or expressions arranged in rows and columns.
    • Elements of a matrix: The numbers, symbols, or expressions present in a matrix.
    • Types of matrices: Square matrix, Row matrix, Column matrix, Zero matrix, Identity matrix, etc.
  • Elementary Row Operations
    • Definition: The three operations that can be performed on a matrix without changing the solutions of the system of linear equations.
    • Three elementary row operations:
      • Interchanging two rows
      • Multiplying a row by a non-zero constant
      • Adding a multiple of one row to another row
  • Row Echelon Form of a Matrix
    • Definition: A matrix is said to be in row echelon form if it satisfies certain conditions.
    • Conditions for a matrix to be in row echelon form:
      • All rows containing only zeros are at the bottom of the matrix.
      • The first non-zero element (leading coefficient) of a row is always to the right of the leading coefficient of the row above it.
      • Any row that does not consist entirely of zeros has a leading coefficient of 1.
  • Example 1: Transforming a matrix into row echelon form
    • Given matrix: [1, 2, -3; 0, 4, 5; 0, 0, 6]
    • Step 1: Interchange row 1 and row 2
    • Step 2: Multiply row 2 by 1/4
    • Step 3: Multiply row 3 by 1/6
    • The resulting matrix is in row echelon form: [1, 2, -3; 0, 1, 5/4; 0, 0, 1]
  • Example 2: Solving a system of linear equations using row echelon form
    • Given system of equations:
      • 2x + y - z = 7
      • x - y + 3z = -1
      • -3x + y + 2z = 3
    • Write the augmented matrix: [2, 1, -1, 7; 1, -1, 3, -1; -3, 1, 2, 3]
    • Convert the matrix to row echelon form
    • Solve for the variables using back substitution
  • Properties of Elementary Row Operations
    • They do not change the solutions of the system of linear equations.
    • They can be used to find the inverse of a matrix.
    • They can be used to solve systems of linear equations more efficiently.
  • Applications of Row Echelon Form
    • Solving systems of linear equations
    • Determining the rank of a matrix
    • Finding the inverse of a matrix
    • Solving homogeneous systems of linear equations

Slide 11: Properties of Elementary Row Operations

  • They do not change the solutions of the system of linear equations.
  • They can be used to find the inverse of a matrix.
  • They can be used to solve systems of linear equations more efficiently.
  • The row operations can be reversed to obtain the original matrix.
  • Performing more than one row operation at a time is equivalent to performing each operation separately.

Slide 12: Applications of Row Echelon Form

  • Solving systems of linear equations:
    • Convert the augmented matrix to row echelon form and solve for the variables.
  • Determining the rank of a matrix:
    • The number of non-zero rows in the row echelon form is equal to the rank of the matrix.
  • Finding the inverse of a matrix:
    • If the row echelon form of a matrix is the identity matrix, then the original matrix is invertible.
  • Solving homogeneous systems of linear equations:
    • If the row echelon form of the augmented matrix has no row with non-zero entries in the last column, the system has infinitely many solutions.

Slide 13: Example 3: Finding the Rank of a Matrix

  • Given matrix: [1, 2, -3; 0, 3, -2; 0, 0, 0]
  • Convert the matrix to row echelon form:
    • Step 1: Multiply row 2 by 1/3
    • Step 2: Multiply row 1 by 1/2
  • The row echelon form is: [1, 2, -3; 0, 1, -2/3; 0, 0, 0]
  • The rank of the matrix is 2 (number of non-zero rows).

Slide 14: Example 4: Finding the Inverse of a Matrix

  • Given matrix: [1, 2; 3, 4]
  • Write the augmented matrix: [1, 2, 1, 0; 3, 4, 0, 1]
  • Convert the matrix to row echelon form:
    • Step 1: Multiply row 2 by -3 and add to row 1
  • The row echelon form is: [1, 2, 1, 0; 0, -2, -3, 1]
  • Multiply row 1 by 1/2 to make the leading coefficient 1
  • Multiply row 2 by -1/2 to make the leading coefficient 1
  • The inverse of the given matrix is: [-2, 1; 3/2, -1/2]

Slide 15: Example 5: Solving Homogeneous Systems of Linear Equations

  • Given system of equations:
    • 2x + 3y - z = 0
    • 4x + y + 2z = 0
    • 2x - y + 3z = 0
  • Write the augmented matrix: [2, 3, -1, 0; 4, 1, 2, 0; 2, -1, 3, 0]
  • Convert the matrix to row echelon form:
    • Step 1: Divide row 1 by 2
    • Step 2: Multiply row 1 by -4 and add to row 2
    • Step 3: Multiply row 1 by -2 and add to row 3
  • The row echelon form is: [1, 3/2, -1/2, 0; 0, -5, -1, 0; 0, 0, 0, 0]
  • The system of equations has infinitely many solutions.

Slide 16: Summary

  • Matrices are rectangular arrays of numbers, symbols, or expressions.
  • Elementary row operations can be performed on matrices without changing the solutions of the system of linear equations.
  • Row echelon form is a specific form of a matrix that satisfies certain conditions.
  • Row echelon form can be used to solve systems of linear equations, determine the rank of a matrix, find the inverse of a matrix, and solve homogeneous systems of linear equations.
  • The properties of elementary row operations allow for efficient manipulation and solving of matrices.
  • Examples provided throughout the lecture illustrate the application of these concepts.

Slide 21: Determinants of Matrices

  • Definition: The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix.
  • Determinant of a 2x2 matrix:
    • det([a, b; c, d]) = ad - bc
  • Determinant of a 3x3 matrix:
    • det([a, b, c; d, e, f; g, h, i]) = aei + bfg + cdh - ceg - bdi - afh
  • Properties of Determinants:
    • If a matrix has a row or column of zeros, its determinant is zero.
    • Interchanging two rows or columns of a matrix changes the sign of its determinant.
    • If all the elements of a row or column of a matrix are multiplied by the same non-zero scalar, the determinant is multiplied by that scalar.
    • The determinant of the transpose of a matrix is equal to the determinant of the original matrix.
    • If two rows or columns of a matrix are identical, its determinant is zero.

Slide 22: Inverse of a Matrix

  • Definition: The inverse of a square matrix A, denoted as A^-1, is the matrix such that A * A^-1 = I, where I is the identity matrix.
  • Conditions for a matrix to have an inverse:
    • The matrix must be square (number of rows = number of columns).
    • The determinant of the matrix must be non-zero.
  • Formula for calculating the inverse of a 2x2 matrix:
    • A^-1 = 1/det(A) * [d, -b; -c, a]
  • Formula for calculating the inverse of a 3x3 matrix:
    • A^-1 = 1/det(A) * [ei - fh, ch - bi, bf - ce; fg - di, ai - cg, cd - af; dh - eg, bg - ah, ae - bd]
  • Properties of Inverse Matrices:
    • If A is invertible, then A^-1 is invertible and (A^-1)^-1 = A.
    • The inverse of the product of two matrices is the product of their inverses in reverse order.
    • The inverse of the transpose of a matrix is equal to the transpose of its inverse.
    • The inverse of a diagonal matrix is a diagonal matrix with the reciprocals of the diagonal elements.
    • The inverse of the identity matrix is the identity matrix itself.

Slide 23: Example 1: Finding the Determinant of a 3x3 Matrix

  • Given matrix: [1, 2, 3; 4, 5, 6; 7, 8, 9]
  • Using the formula for a 3x3 matrix:
    • det([1, 2, 3; 4, 5, 6; 7, 8, 9]) = 1*(5*9 - 6*8) - 2*(4*9 - 6*7) + 3*(4*8 - 5*7)
    • det([1, 2, 3; 4, 5, 6; 7, 8, 9]) = 1*(45 - 48) - 2*(36 - 42) + 3*(32 - 35)
    • det([1, 2, 3; 4, 5, 6; 7, 8, 9]) = -3 + 12 - 9 = 0

Slide 24: Example 2: Finding the Inverse of a 2x2 Matrix

  • Given matrix: [3, 2; 1, 4]
  • Using the formula for a 2x2 matrix:
    • A^-1 = 1/det(A) * [d, -b; -c, a] = 1/(3*4 - 2*1) * [4, -2; -1, 3]
    • A^-1 = 1/10 * [4, -2; -1, 3] = [2/5, -1/5; -1/10, 3/10]

Slide 25: Example 3: Finding the Inverse of a 3x3 Matrix

  • Given matrix: [2, -1, 3; 1, 2, -1; 4, 3, 2]
  • Using the formula for a 3x3 matrix:
    • A^-1 = 1/det(A) * [ei - fh, ch - bi, bf - ce; fg - di, ai - cg, cd - af; dh - eg, bg - ah, ae - bd]
    • A^-1 = 1/(2*(-1*2-3*3) - (-1)*(-1*2-4*3) + 3*(3*2-4*2)) * [-1*2 - (-1)*3, (-1)*(-1) - 3*(-1), (-1)*4 - (-1)*3; (-1)*2 - 3*3, 2*(-1) - 4*(-1), 4*(-1) - (-1)*2; 3*2 - 4*2, 2*4 - 4*(-1), 2*(-1) - (-1)*2]
    • A^-1 = 1/10 * [2 - 3, 1 - (-3), -4 + 3; -2 - 9, 2 + 4, -4 - 2; 6 - 8, 8 + 4, -2 - 2]
    • A^-1 = 1/10 * [-1, 4, -1; -11, 6, -6; -2, 12, -4]

Slide 26: Example 4: Properties of Determinants

  • Given matrix A: [2, 1; 3, 4] and matrix B: [1, -1; 2, -2]
  • Calculate the determinants of A and B:
    • det(A) = 2*4 - 1*3 = 8 - 3 = 5
    • det(B) = 1*(-2) - (-1)*2 = -2 - (-2) = 0
  • Check the properties of determinants:
    • A has non-zero determinant, so it has an inverse.
    • B has determinant equal to zero, so it does not have an inverse.
    • Interchanging rows of A changes the sign of its determinant.
    • Multiplying a row of A by a non-zero scalar multiplies its determinant by that scalar.

Slide 27: Example 5: Properties of Inverse Matrices

  • Given matrix A: [1, 2; 3, 4] and its inverse A^-1: [-2, 1; 3/2, -1/2]
  • Calculate the product of A and A^-1:
    • A * A^-1 = [1, 2; 3, 4] * [-2, 1; 3/2, -1/2]
    • A * A^-1 = [(-2) + (3), (1) + (-1); (3*3/2) + (4*(-1/2)), (3/2) + (-2/2)]
    • A * A^-1 = [1, 0; 0, 1]
  • Check the properties of inverse matrices:
    • (A^-1)^-1 = A, so the inverse of the inverse is the original matrix.
    • A * A^-1 = I, so the product of a matrix and its inverse is the identity matrix.

Slide 28: Solving Systems of Linear Equations using Matrices

  • Given system of equations:
    • 2x + 3y = 8
    • 4x + 5y = 13
  • Write the augmented matrix: [2, 3, 8; 4, 5, 13]
  • Convert the matrix to row echelon form:
    • Step 1: Divide row 1 by 2
    • Step 2: Multiply row 1 by -4 and add to row 2
  • The row echelon form is: [1, 3/2, 4; 0, 1, 1]
  • Solve for the variables using back substitution:
    • y = 1
    • Substitute y = 1 into the first equation: 2x + 3*1 = 8
    • Solve for x: 2x + 3 = 8, 2x = 5, x = 5/2

Slide 29: Determining the Rank of a Matrix

  • Definition: The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.
  • Steps to determine the rank of a matrix:
    1. Convert the matrix to row echelon form.
    2. Count the number of non-zero rows in the row echelon form.
    3. The rank of the matrix is equal to the number of non-zero rows.
  • Example:
    • Given matrix: [1, 2, 3; 4, 5, 6; 7, 8, 9]
    • Convert the matrix to row echelon form: [1, 2, 3; 0, -3, -6; 0, 0, 0]
    • The row echelon form has 2 non-zero rows.
    • The rank of the matrix is 2.

Slide 30: Solving Homogeneous Systems of Linear Equations

  • Definition: A homogeneous system of linear equations is a system in which all the constants on the right-hand side of the equations are zero.
  • Method to solve a homogeneous system of linear equations:
    1. Write the augmented matrix of the system.
    2. Convert the augmented matrix to row echelon form.
    3. Count the number of non-zero rows in the row echelon form.
    4. If the number of non-zero rows is equal to the number of variables, the system has a unique solution where all variables are zero.
    5. If the number of non-zero rows is less than the number of variables, the system has infinitely many solutions.