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7.7B-Minors and Cofactors

7.7B-Minors and Cofactors. Minors of entry 1. Eliminate the row i and column j the entry is in 2. Find determinant of resulting matrix 2x2: 3x3: diagonal method OR expand cofactors Larger square matrix: expand cofactors Cofactor,

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7.7B-Minors and Cofactors

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  1. 7.7B-Minors and Cofactors • Minors of entry 1. Eliminate the row i and column j the entry is in 2. Find determinant of resulting matrix • 2x2: • 3x3: diagonal method OR expand cofactors • Larger square matrix: expand cofactors • Cofactor, = (sign pattern for entry )(Minor of entry, ) Sign patterns start + in and alternate + = keepsign of that entry in original matrix - = use opposite sign of that entry in original matrix

  2. Examples: Find the Minors and Cofactor for the given entries • [A]= • 1. • b) • 2. • a) • b)

  3. Finding Determinant with Expansion of Cofactors (any square size) • 1. Choose a row OR column to expand. • The more zeros in the row or column, the better. • 2. Find the Minor for EACH entry in the row or column chosen. • 3. Identify the sign pattern for the row or column chosen • 4. Find the cofactor for each entry in the row or column chosen • 5. Repeat process for EACH Minor if larger than 3x3 • 6. Determinant = SUM of the cofactors of the chosen row or column.

  4. Example: Find the Determinant by Expanding Cofactors • 3. • A) Expand Row 1 B) Expand Column 2

  5. Example: Find Determinant by Expanding Cofactors • 4. 8 6 0 2

  6. Determinant with Diagonal Method • Diagonal Method • ONLY for 3x3 • Repeat first 2 columns • Multiply 3 entries diagonally down 3 times (beginning with ), separated by addition • Multiply 3 entries diagonally up 3 times (beginning with ), separated by subtraction. • mult. + mult. + mult. – mult. – mult. – mult

  7. Example: Find the Determinant using the Diagonal Method • 5.

  8. Triangular Matrices • Upper Triangular: Main diagonal & above are numbers (Below = all ZEROS) • Lower Triangular: Main diagonal & below are numbers (Above = all ZEROS) • Diagonal: Main diagonal is numbers (Above AND Below = all ZEROS) • Determinant of ALL triangular & diagonal matrices = product (multiply) of ALL #’s on main diagonal. • 6. 7. 8. • upper triangular lower triangular diagonal • det= det= det= • Assignment: 7.7B p. 556 # 10 (diagonal method), 12 (triangular), 20-26 even (expand cofactors method), 30 & 32 (triangular),36,40

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