Sec 3.6 Determinants

1
Sec 3.6 Determinants
2x2 matrix
Evaluate the determinant of
2
Sec 3.6 Determinants
Cramers Rule (solve linear system)
Solve the system
3
Sec 3.6 Determinants
Cramers Rule (solve linear system)
Solve the system
4
Sec 3.6 Determinants
Def Minors
Let A aij be an nxn matrix . The ijth minor
of A ( or the minor of aij) is the determinant
Mij of the (n-1)x(n-1) submatrix after you
delete the ith row and the jth column of A.
Find
5
Sec 3.6 Determinants
Def Cofactors
Let A aij be an nxn matrix . The ijth
cofactor of A ( or the cofactor of aij) is
defined to be
Find
signs
6
Sec 3.6 Determinants
3x3 matrix
Find det A
7
Sec 3.6 Determinants
The cofactor expansion of det A along the first
row of A

• Note
• 3x3 determinant expressed in terms of three
  2x2 determinants
• 4x4 determinant expressed in terms of four
  3x3 determinants
• 5x5 determinant expressed in terms of five
  4x4 determinants
• nxn determinant expressed in terms of n
  determinants of size (n-1)x(n-1)

8
Sec 3.6 Determinants
nxn matrix
We multiply each element by its cofactor ( in
the first row)
Also we can choose any row or column
Th1 the det of an nxn matrix can be obtained by
expansion along any row or column.
i-th row
j-th row
9
Row and Column Properties
Prop 1 interchanging two rows (or columns)
10
Row and Column Properties
Prop 2 two rows (or columns) are identical
11
Row and Column Properties
Prop 3 (k) i-th row j-th row
(k) i-th col j-th col
12
Row and Column Properties
Prop 4 (k) i-th row
(k) i-th col
13
Row and Column Properties
Prop 5 i-th row B i-th row A1 i-th
row A2
Prop 5 i-th col B i-th col A1 i-th
col A2
14
Row and Column Properties
Either upper or lower
Zeros below main diagonal
Zeros above main diagonal
Prop 6 det( triangular ) product of
diagonal
15
Row and Column Properties
16
Transpose
Prop 6 det( matrix ) det( transpose)
17
Transpose
18
Determinant and invertibility
THM 2 The nxn matrix A is invertible
detA 0
19
Determinant and inevitability
THM 2 det ( A B ) det A
det B
Note
Proof
Example compute
20
Cramers Rule (solve linear system)
Solve the system
21
Cramers Rule (solve linear system)
Use cramers rule to solve the system
22
Adjoint matrix
Def Cofactor matrix
Let A aij be an nxn matrix . The cofactor
matrix Aij
signs
Find the cofactor matrix
Find the adjoint matrix
Def Adjoint matrix of A
23
Another method to find the inverse
Thm2 The inverse of A
Find the inverse of A
24
Computational Efficiency
The amount of labor required to compute a
numerical calculation is measured by the number
of arithmetical operations it involves
Goal let us count just the number of
multiplications required to evaluate an nxn
determinant using cofactor expansion
2x2 2 multiplications
3x3 three 2x2 determinants ? 3x2 6
multiplications
4x4 four 3x3 determinants ? 4x3x2 24
multiplications
5x5 four 3x3 determinants ? 4x3x2 24
multiplications
- - - - - - - - - - - - - - - - - - - -
- - - - - - - -
nxn n (n-1)x(n-1) determinants ? nxx3x2
n! multiplications
25
Computational Efficiency
Goal let us count just the number of
multiplications required to evaluate an nxn
determinant using cofactor expansion
nxn determinants ? requires n!
multiplications
a typical 1998 desktop computer , using MATLAB
and performing aonly 40 million operations per
second
To evaluate a determinant of a 15x15 matrix using
cofactor expansion ? requires
a supercomputer capable of a billion operations
per seconds
To evaluate a detrminant of a 25x25 matrix
using cofactor expansion ? requires
26
Monday Quiz