Matrix and Determinant - Properties of determinants

Slide 1:

  • Introduce the topic: Properties of determinants
  • Define determinant and its significance in matrix algebra
  • Briefly mention that determinants have several important properties
  • State the goal of the lecture: To understand and apply the properties of determinants

Slide 2:

  • Property 1: Multiplying a row (or column) by a scalar
    • Equation: If A is a matrix and k is a scalar, then |kA| = k^n|A|, where n is the order of the matrix

Slide 3:

  • Property 2: Interchanging two rows (or columns)
    • Equation: If A is a matrix, then |A| = -|A'|

Slide 4:

  • Property 3: Adding a multiple of one row (or column) to another row (or column)
    • Equation: If A is a matrix and R_i and R_j are two rows, then |A| = |A’+R_j| = |A’+kR_j| = …

Slide 5:

  • Property 4: If two rows (or columns) are proportional, then the determinant is zero
    • Equation: If A is a matrix and R_i and R_j are proportional, then |A| = 0

Slide 6:

  • Property 5: If a matrix has a row (or column) with all elements zero, then the determinant is zero
    • Equation: If A is a matrix and R_i contains all zeroes, then |A| = 0

Slide 7:

  • Property 6: If a matrix is triangular, then the determinant is the product of diagonal elements
    • Equation: If A is a triangular matrix, then |A| = a_11 * a_22 * a_33 * … * a_nn

Slide 8:

  • Property 7: If a matrix is invertible (non-singular), then its determinant is non-zero
    • Equation: If A is an invertible matrix, then |A| ≠ 0

Slide 9:

  • Property 8: The determinant of the product of two matrices is the product of their determinants
    • Equation: If A and B are matrices, then |AB| = |A| * |B|

Slide 10:

  • Recap of the properties covered so far
  • Emphasize the importance of understanding and applying these properties in solving problems related to determinants

Slide 11:

  • Property 9: If two rows (or columns) of a matrix are interchanged, then the determinant changes its sign
    • Equation: If A is a matrix and R_i and R_j are interchanged, then |A’| = -|A|

Slide 12:

  • Property 10: If any two rows (or columns) of a matrix are identical, then the determinant is zero
    • Equation: If A is a matrix and R_i and R_j are identical, then |A| = 0

Slide 13:

  • Property 11: If each element of a row (or column) is expressed as the sum or difference of two terms, then the determinant can be expressed as the sum of two determinants
    • Equation: If A is a matrix and R_i = R_x + R_y, then |A| = |A’+R_x| + |A’+R_y|

Slide 14:

  • Property 12: If a constant multiple of one row (or column) is added to another row (or column), then the determinant remains the same
    • Equation: If A is a matrix and R_i + kR_j = R_x, then |A| = |A’+R_x|

Slide 15:

  • Property 13: If all elements of a row (or column) are multiplied by a non-zero scalar, then the determinant is multiplied by the same scalar
    • Equation: If A is a matrix and kR_i = R_x, then |A| = k|A’+R_x|

Slide 16:

  • Property 14: If a constant multiple of one row (or column) is subtracted from another row (or column), then the determinant remains the same
    • Equation: If A is a matrix and R_i - kR_j = R_x, then |A| = |A’+R_x|

Slide 17:

  • Property 15: Changing the sign of a row (or column) changes the sign of the determinant
    • Equation: If A is a matrix and -R_i = R_x, then |A| = -|A’+R_x|

Slide 18:

  • Property 16: If each element of a row (or column) is expressed as the sum or difference of two terms, and all the other elements of the rows (or columns) are zero, then the determinant is the sum or difference of two determinants
    • Equation: If A is a matrix and a_1j = b_1j + c_1j and a_ij = 0 for i ≠ 1, then |A| = |A’| = |B’| + |C'|

Slide 19:

  • Property 17: If A and B are two matrices of the same order, then |A+B| ≠ |A| + |B|
    • Equation: If A and B are matrices, then |A+B| ≠ |A| + |B|

Slide 20:

  • Property 18: If A is a matrix obtained by multiplying a row (or column) of B by k, then |A| = k|B|
    • Equation: If A is a matrix and B is a matrix obtained by multiplying R_i of A by k, then |A| = k|B|

Slide 21:

  • Property 19: If A and B are two matrices such that all corresponding elements are equal, then |A| = |B|
    • Equation: If A and B are matrices with a_ij = b_ij for all i and j, then |A| = |B|

Slide 22:

  • Property 20: If A is a square matrix and k is a non-zero scalar, then |kA| = k^n|A|, where n is the order of the matrix
    • Equation: If A is a matrix and k is a non-zero scalar, then |kA| = k^n|A|

Slide 23:

  • Property 21: If A is a square matrix and k is a scalar, then |A^T| = |A|
    • Equation: If A is a matrix, then |A^T| = |A|

Slide 24:

  • Property 22: If A is a square matrix and all its diagonal elements are non-zero, then A is invertible (non-singular)
    • Equation: If A is a matrix and all diagonal elements (a_ii) are non-zero, then A is invertible

Slide 25:

  • Property 23: If A is a square matrix and has a row (or column) of all zeroes, then |A| = 0
    • Equation: If A is a matrix and has a row (or column) of all zeroes, then |A| = 0

Slide 26:

  • Property 24: If A is a square matrix and A^T is its transpose, then |A^T| = |A|
    • Equation: If A is a matrix, then |A^T| = |A|

Slide 27:

  • Property 25: If A is a square matrix and its transpose, A^T, is a skew-symmetric matrix, then |A| = 0
    • Equation: If A is a skew-symmetric matrix, then |A| = 0

Slide 28:

  • Property 26: If A is a square matrix and its transpose, A^T, is a symmetric matrix, then |A| ≠ 0
    • Equation: If A is a symmetric matrix, then |A| ≠ 0

Slide 29:

  • Recap of all the properties covered in the lecture
  • Emphasize the importance of understanding these properties for solving problems related to determinants

Slide 30:

  • Summary and conclusion
  • Encourage further practice and exploration of the properties of determinants
  • Thank the students for their attention and conclude the lecture