BAI CM20144 Applications I: Mathematics for Applications Mark Wood

BAI Determinants Evaluation Methods Properties Examples Test 5 Todays Tutorial

BAI Evaluating Determinants 1

BAI Diagonals Method Only works for 2 x 2 and 3 x 3 Multiply forward diagonal elements and add Multiply backward diagonal elements and subtract Evaluating Determinants 1

BAI Diagonals Method Only works for 2 x 2 and 3 x 3 Multiply forward diagonal elements and add Multiply backward diagonal elements and subtract Cofactor Method Pick the row or column with the most zeros Calculate the cofactor for each element and sum Cofactor = sign x minor Signs alternate Minor = determinant of remaining matrix… Evaluating Determinants 1

BAI Diagonals Method Only works for 2 x 2 and 3 x 3 Multiply forward diagonal elements and add Multiply backward diagonal elements and subtract Cofactor Method Pick the row or column with the most zeros Calculate the cofactor for each element and sum Cofactor = sign x minor Signs alternate Minor = determinant of remaining matrix… Recursive Evaluating Determinants 1

BAI Example: Diagonals

BAI Example: Cofactors

BAI Properties of Determinants

BAI Singular Matrices Determinant = 0 (otherwise nonsingular) Row or column of zeros singular Two rows proportional singular Properties of Determinants

BAI Singular Matrices Determinant = 0 (otherwise nonsingular) Row or column of zeros singular Two rows proprtional singular Invertible nonsingular Properties of Determinants

BAI Singular Matrices Determinant = 0 (otherwise nonsingular) Row or column of zeros singular Two rows proprtional singular Invertible nonsingular Other properties Scalar multiple: |cA| = c n |A|(n = matrix dim) Product: |AB| = |A||B| Transpose: |A t | = |A| Inverse: |A -1 | = 1/|A|(if A -1 exists) Properties of Determinants

BAI A and B are 3 x 3 matrices |A| = -3, |B| = 2 Calculate: |AB| |AA t | |A t B| |3A 2 B| |2AB -1 | |(A 2 B -1 ) t | Example: Properties of Determinants

BAI Evaluating Determinants 2

BAI Row Operations and Determinants 1) Multiply by c c|A| 2) Swap two rows -|A| 3) Add multiple of one row to another |A| Evaluating Determinants 2

BAI Row Operations and Determinants 1) Multiply by c c|A| 2) Swap two rows -|A| 3) Add multiple of one row to another |A| Get zero columns / rows and use cofactors Evaluating Determinants 2

BAI Row Operations and Determinants 1) Multiply by c c|A| 2) Swap two rows -|A| 3) Add multiple of one row to another |A| Get zero columns / rows and use cofactors Numerical Method Use row ops to get matrix into upper triangular form Only need 2) and 3) Keep track of op 2) Determinant is product of diagonal elements Zero on diagonal & zeros below singular Evaluating Determinants 2

BAI 1 0 – – –1 0 Example: Numerical Evaluation

BAI Example: Numerical Evaluation

BAI Other Stuff?

BAI A -1 = adj(A) / |A| Adjoint is transpose of matrix of cofactors Other Stuff?

BAI A -1 = adj(A) / |A| Adjoint is transpose of matrix of cofactors System of Equations AX = B Unique solution A nonsingular Otherwise, could be many or no solutions Cramers Rule: x i = |A i | / |A| Other Stuff?