OCE301 Part II: Linear Algebra lecture 3

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OCE301 Part II Linear Algebralecture 3
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Determinants

• Only square matrices possess determinants

• det A is a number and has no elements.

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Determinant of a Low Order Matrix
first order
second order
third order
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Diagonal Method for evaluating determinant
second-order
Fourth- and higher-order determinants may not
be evaluated by following the same procedures.

third-order
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Third-Order Determinants
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Minors and Cofactors
(are referring to determinant, not matrix)
the minor associated with a12 is denoted as M12
deleting first row and second column
Cofactor (is the signed minor)
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General Definition for Determinant
for any row j
for any column k
(to accept this definition) it must be shown that
the same expansion is obtained no matter which
row or column is selected
(using inductive method)
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Matlab Example Determinant
a1 2 5 2 4 9 3 4 6 a 1 2 5
2 4 9 3 4 6
ddet(a) d -2
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Properties of Determinants
a1 2 5 2 4 9 3 4 6 b2 3 4 5 7 6 2 1
3 detadet(a) deta -2 detbdet(b) detb
-15 detabdet(ab) detab 30
b2 3 4 5 7 6 2 1 3 detbpdet(b') detbp
-15 detbdet(b) detb -15
det B det BT
(det A) (det B) det (AB)
det (BA) det (AB)
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Determinant Behavior under Row Operations
Row operations for matrices
Change on determinant
Interchange of two rows
multiply (-1)
Addition of a constant multiple of one row to
another row
no change
Multiplication of a row by a nonzero constant c
multiply c
(rigorous proof shown in page 345)
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Interchange of Two Rows Matlab Demo
a1 2 3 2 3 5 3 5 7 b2 3 5 1 2
3 3 5 7 c3 5 7 2 3 5 1 2
3 det_adet(a) det_bdet(b) det_cdet(c)
det_a 1 det_b -1 det_c -1
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Simultaneous Equations in Matrix Form
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Two Simultaneous Equations
_
the denominators can be represented by
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Cramers Rule Two Equations
the numerators can be represented by
Cramers rule
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Three Simultaneous Equations
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Cramers Rule Three Equations
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Cramers Rule General
Dk is the determinant obtained from D by
replacing in D the kth column by the column with
the entries b1, , bn.
Cramers rule is not practical in computations,
but is of theoretical interest
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Inverse of a Matrix
A-1 the inverse of A
AA-1 A-1A I (for square matrix only)
If A has no inverse, then A is called a singular
matrix. If A has an inverse, then A is called a
nonsingular matrix.
If A has an inverse, the inverse is unique.
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Inverse/Rank/Determinant
The inverse A-1 of an n-by-n matrix A exists (A
is nonsingular) if and only if rank(A) n,
hence if and only if det A is not equal to
zero.
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Determination of the Inverse(Gauss-Jordan
Elimination)
X A-1
AX I
all A, X and I are n-by-n matrices
augmented matrix
further row operations
Gauss elimination
Gauss-Jordan elimination
U upper triangular
I X K
I X X A-1
K A-1
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Matlab Inverse
A-1 1 2 3 -1 1 -1 3 4 A -1 1
2 3 -1 1 -1 3 4 A_inv
inv(A) A_inv -0.7000 0.2000 0.3000
-1.3000 -0.2000 0.7000 0.8000 0.2000
-0.2000
AA-1 A-1A I
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Inverse of a Diagonal Matrix
a1 0 0 0 2 0 0 0 4 a 1 0 0
0 2 0 0 0 4
ainvinv(a) ainv 1.0000 0
0 0 0.5000 0
0 0 0.2500
A-1 is diagonal with entries 1/a11, , 1/ann
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Some Useful Formulas for Inverses
(AC) -1 C-1 A-1
(A -1 )-1 A
page 353, a general form for an n-by-n matrix.
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Homework Assignment

• Problem Set 6.6 (page 349) problems 6,11
• Problem Set 6.7 (page 357) problems 4,12
• Problem Set 7.1 (page 375) problems 7,12
• Problem Set 7.3 (page 384) problem 6
• Problem Set 7.4 (page 390) problem 6

You need to work on each problem by hand
(providing details), then use MATLAB to check
your solutions. Hand in BOTH hand calculations
and MATLAB operations.