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BAI
CM20144
Applications I:
Mathematics for Applications
Mark Wood
http://www.cs.bath.ac.uk/~cspmaw
BAI
• Determinants
Evaluation Methods
Properties
Examples
• Test 5
Today’s Tutorial
BAIEvaluating 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
3 1 4
-7 -2 1
9 1 -1
Example: Diagonals
BAI
3 1 4
-7 -2 1
9 1 -1
Example: Cofactors
BAIProperties 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| = cn|A| (n = matrix dim)
Product: |AB| = |A||B|
Transpose: |At| = |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|
|AAt|
|AtB|
|3A2B|
|2AB-1|
|(A2B-1)t|
Example: Properties of Determinants
BAIEvaluating 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 –2 1
2 1 0 2
-1 1 –2 1
3 1 –1 0
Example: Numerical Evaluation
BAI
1 -1 0 2
-1 1 0 0
2 -2 0 1
3 1 5 -1
Example: Numerical Evaluation
BAIOther 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
Cramer’s Rule: xi = |Ai| / |A|
Other Stuff?
