Determinants - Expansion of determinant is not restricted

• Expansion of determinants can be done along any row or column.
• It involves multiplying each element of a row or column by its corresponding cofactor and summing up the products.
• The value obtained is the value of the determinant.

Example 1:

Consider the matrix A: | a | b | ||| | c | d |

Expansion along the first row:

Determinant of A = (a)(cofactor of a) + (b)(cofactor of b)

Expansion along the second column:

Determinant of A = (b)(cofactor of b) + (d)(cofactor of d)

Example 2:

Consider the matrix B: | 2 | 5 | 1 | |||| | 7 | 3 | 4 | | 0 | 6 | 8 |

Expansion along the first row:

Determinant of B = (2)(cofactor of 2) + (5)(cofactor of 5) + (1)(cofactor of 1)

Expansion along the second column:

Determinant of B = (5)(cofactor of 5) + (3)(cofactor of 3) + (6)(cofactor of 6)

Expansion along the third row:

Determinant of B = (0)(cofactor of 0) + (6)(cofactor of 6) + (8)(cofactor of 8)

Equations involving determinants:

• The determinant of a 2x2 matrix can be calculated using the equation: | a | b | ||| | c | d |

  Determinant = (a)(d) - (b)(c)

• The determinant of a 3x3 matrix can be calculated using the equation:

  a1	b1	c1
  a2	b2	c2
  a3	b3	c3

  Determinant = (a1)(cofactor of a1) + (b1)(cofactor of b1) + (c1)(cofactor of c1)

• The determinant of a 4x4 matrix can be calculated using the equation:

  a1	b1	c1	d1
  a2	b2	c2	d2
  a3	b3	c3	d3
  a4	b4	c4	d4

  Determinant = (a1)(cofactor of a1) + (b1)(cofactor of b1) + (c1)(cofactor of c1) + (d1)(cofactor of d1)

Expansion of a determinant with zeros:

If a determinant has a row or column consisting entirely of zeros, its expansion can be simplified by considering the zeros.

• If a row or column of a determinant is all zeros, then the value of the determinant is zero.

Example 3:

Consider the matrix C: | 2 | 0 | 3 | |||| | 0 | 0 | 0 | | 1 | 0 | 4 | Since the second row consists entirely of zeros, the determinant is zero.

Slide 11

• Determinants can also be expanded using minors and cofactors.
• The minor of an element in a matrix is the determinant of the matrix formed by deleting the row and column which contain that element.
• The cofactor of an element is the product of its minor and a sign, which depends on the position of the element in the matrix.
• The sign of the cofactor is positive if the sum of the row number and column number is even, and negative if the sum is odd.

Slide 12

• The expansion of a determinant using minors and cofactors involves multiplying each element of a row or column by its corresponding cofactor and summing up the products.
• The value obtained is the value of the determinant.
• This method is often used for expanding determinants with more than 3 rows or columns.

Slide 13

• Example: Consider the matrix D: | 1 | 2 | 3 | |||| | 4 | 5 | 6 | | 7 | 8 | 9 |
• Expansion along the first row: Determinant of D = (1)(cofactor of 1) + (2)(cofactor of 2) + (3)(cofactor of 3)

Slide 14

• Example: Consider the matrix E: | 2 | -1 | 3 | ||-|| | 0 | 5 | 2 | | 1 | 3 | 4 |
• Expansion along the second column: Determinant of E = (1)(cofactor of 1) + (5)(cofactor of 5) + (3)(cofactor of 3)

Slide 15

• Example: Consider the matrix F:

  3	7	2
  -1	-4	-2
  0	-6	8

• Expansion along the third row: Determinant of F = (0)(cofactor of 0) + (-6)(cofactor of -6) + (8)(cofactor of 8)

Slide 16

• Determinants satisfy certain properties.
• Property 1: The value of a determinant remains unchanged if the rows and columns are interchanged.
• Property 2: The value of a determinant remains unchanged if the elements in any row or column are multiplied by the same non-zero constant.
• Property 3: If any two rows or columns of a determinant are the same, then the determinant is zero.

Slide 17

• Property 4: If all the elements of a row or column of a determinant are zero, then the value of the determinant is zero.
• Property 5: If two rows or columns of a determinant are identical except for sign, then the determinant is zero.
• Property 6: The value of a determinant is unchanged if a multiple of any row or column is added to another row or column.

Slide 18

• Example: Consider the matrix G: | 2 | 3 | 4 | |||| | 5 | 6 | 7 | | 8 | 9 | 10 |
• Property application:
  • Multiply the third row by 2 and add it to the second row.

Slide 19

• Inverse of a matrix:

  • A square matrix A has an inverse if its determinant is non-zero.
  • The matrix A^(-1) is called the inverse of A, and it satisfies the property AA^(-1) = A^(-1)A = I, where I is the identity matrix.

• The inverse of a 2x2 matrix can be calculated using the equation: | a | b | ||| | c | d |

  A^(-1) = (1/det(A)) * | d | -b | | -c | a |

Slide 20

• The inverse of a 3x3 matrix can be calculated using the equation:

  a1	b1	c1
  a2	b2	c2
  a3	b3	c3

  A^(-1) = (1/det(A)) * | b2c3 - b3c2 | a3c2 - a2c3 | a2b3 - a3b2 | | b3c1 - b1c3 | a1c3 - a3c1 | a3b1 - a1b3 | | b1c2 - b2c1 | a2c1 - a1c2 | a1b2 - a2b1 |

Determinants - Expansion of determinant is not restricted

• Expansion of determinants can be done along any row or column.

• It involves multiplying each element of a row or column by its corresponding cofactor and summing up the products.

• The value obtained is the value of the determinant. Example 1: Consider the matrix A: | a | b | ||| | c | d |

• Expansion along the first row: Determinant of A = (a)(cofactor of a) + (b)(cofactor of b)

• Expansion along the second column: Determinant of A = (b)(cofactor of b) + (d)(cofactor of d) Example 2: Consider the matrix B: | 2 | 5 | 1 | |||| | 7 | 3 | 4 | | 0 | 6 | 8 |

• Expansion along the first row: Determinant of B = (2)(cofactor of 2) + (5)(cofactor of 5) + (1)(cofactor of 1)

• Expansion along the second column: Determinant of B = (5)(cofactor of 5) + (3)(cofactor of 3) + (6)(cofactor of 6)

• Expansion along the third row: Determinant of B = (0)(cofactor of 0) + (6)(cofactor of 6) + (8)(cofactor of 8) Equations involving determinants:

• The determinant of a 2x2 matrix can be calculated using the equation: | a | b | ||| | c | d |

  Determinant = (a)(d) - (b)(c)

• The determinant of a 3x3 matrix can be calculated using the equation:

  a1	b1	c1
  a2	b2	c2
  a3	b3	c3

  Determinant = (a1)(cofactor of a1) + (b1)(cofactor of b1) + (c1)(cofactor of c1)

• The determinant of a 4x4 matrix can be calculated using the equation:

  a1	b1	c1	d1
  a2	b2	c2	d2
  a3	b3	c3	d3
  a4	b4	c4	d4

  Determinant = (a1)(cofactor of a1) + (b1)(cofactor of b1) + (c1)(cofactor of c1) + (d1)(cofactor of d1) Expansion of a determinant with zeros:

• If a determinant has a row or column consisting entirely of zeros, its expansion can be simplified by considering the zeros.

• If a row or column of a determinant is all zeros, then the value of the determinant is zero. Example 3: Consider the matrix C: | 2 | 0 | 3 | |||| | 0 | 0 | 0 | | 1 | 0 | 4 | Since the second row consists entirely of zeros, the determinant is zero.

Determinants - Minor and Cofactor Expansion

• Determinants can also be expanded using minors and cofactors.
• The minor of an element in a matrix is the determinant of the matrix formed by deleting the row and column which contain that element.
• The cofactor of an element is the product of its minor and a sign, which depends on the position of the element in the matrix.
• The sign of the cofactor is positive if the sum of the row number and column number is even, and negative if the sum is odd.
• The expansion of a determinant using minors and cofactors involves multiplying each element of a row or column by its corresponding cofactor and summing up the products.
• The value obtained is the value of the determinant.
• This method is often used for expanding determinants with more than 3 rows or columns. Example: Consider the matrix D: | 1 | 2 | 3 | |||| | 4 | 5 | 6 | | 7 | 8 | 9 |
• Expansion along the first row: Determinant of D = (1)(cofactor of 1) + (2)(cofactor of 2) + (3)(cofactor of 3) Example: Consider the matrix E: | 2 | -1 | 3 | ||-|| | 0 | 5 | 2 | | 1 | 3 | 4 |
• Expansion along the second column: Determinant of E = (1)(cofactor of 1) + (5)(cofactor of 5) + (3)(cofactor of 3) Example: Consider the matrix F:

  3	7	2
  -1	-4	-2
  0	-6	8

• Expansion along the third row: Determinant of F = (0)(cofactor of 0) + (-6)(cofactor of -6) + (8)(cofactor of 8)

Properties of Determinants

• Determinants satisfy certain properties.
• Property 1: The value of a determinant remains unchanged if the rows and columns are interchanged.
• Property 2: The value of a determinant remains unchanged if the elements in any row or column are multiplied by the same non-zero constant.
• Property 3: If any two rows or columns of a determinant are the same, then the determinant is zero. Property 4: If all the elements of a row or column of a determinant are zero, then the value of the determinant is zero. Property 5: If two rows or columns of a determinant are identical except for sign, then the determinant is zero. Property 6: The value of a determinant is unchanged if a multiple of any row or column is added to another row or column. Example: Consider the matrix G: | 2 | 3 | 4 | |||| | 5 | 6 | 7 | | 8 | 9 | 10 |
• Property application: Multiply the third row by 2 and add it to the second row. Inverse of a matrix:
• A square matrix A has an inverse if its determinant is non-zero.
• The matrix A^(-1) is called the inverse of A, and it satisfies the property AA^(-1) = A^(-1)A = I, where I is the identity matrix. The inverse of a 2x2 matrix can be calculated using the equation: | a | b | ||| | c | d | A^(-1) = (1/det(A)) * | d | -b | | -c | a | The inverse of a 3x3 matrix can be calculated using the equation:

  a1	b1	c1
  a2	b2	c2
  a3	b3	c3
  A^(-1) = (1/det(A)) *	b2c3 - b3c2	a3c2 - a2c3
  b3c1 - b1c3	a1c3 - a3c1	a3b1 - a1b3
  b1c2 - b2c1	a2c1 - a1c2	a1b2 - a2b1