or
How can we prove this property?
60
5.7 Multiplicative Inverse of a Square Matrix
61
5.7 Multiplicative Inverse of a Square Matrix
The Cofactor Matrix Let A (aij)nxn be a
square matrix. The cofactor matrix of A, denoted
by cof A, is defined by cof A (Aij)nxn, where
Aij is the cofactor of aij, for every i, j 1,
2,,n.
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5.7 Multiplicative Inverse of a Square Matrix
The Adjoint Matrix Let A be a square matrix.
The transpose of the cofactor matrix of A, i.e.
(cof A)T, is called the adjoint matrix of A,
denoted by adj A.
63
5.7 Multiplicative Inverse of a Square Matrix
The Non-singular/Invertible Matrix A square
matrix A of order n is said to be non-singular or
invertible if and only if there exists a square
matrix B such that AB
BA In, where In is an identity matrix of order
n, and the matrix B is called the multiplicative
inverse or simply the inverse of A, which is
denoted by A-1, i.e. AA-1 A-1A In.
The inverse of a non-singular matrix is unique.
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5.7 Multiplicative Inverse of a Square Matrix
Lemma For any square matrix of order n,
A(adj A) (adj A)A (det A)In, where In is an
identity matrix of order n.
?
Why does (adjA)A equal (detA)In?
65
5.7 Multiplicative Inverse of a Square Matrix
(adj A)A (det A)In
Verification
Why do the elements arrange in this way?
66
5.7 Multiplicative Inverse of a Square Matrix
How can we simplify these three elements?
Then how about the other elements?
Why does each row equal zero?
And
67
5.7 Multiplicative Inverse of a Square Matrix
68
5.7 Multiplicative Inverse of a Square Matrix
(adj A)A (det A)In
Verification
69
5.7 Multiplicative Inverse of a Square Matrix
Lemma For any square matrix of order n,
A(adj A) (adj A)A (det A)In, where In is an
identity matrix of order n.
(adj A)A (det A)AA-1
70
5.7 Multiplicative Inverse of a Square Matrix
71
5.7 Multiplicative Inverse of a Square Matrix
On the other hand,
A square matrix A is said to be singular or not
invertible if and only if the inverse, A-1, of A
does not exist.
A square matrix A is singular if and only if det
A 0.
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5.8 Properties of Inverses
If A and B are non-singular matrices of the same
order, and any scalar ? we have
(1) If AC1 AC2 or C1A C2A, then C1 C2
(2) A-1 is non-singular and (A-1)-1 A
(3) AB is non-singular and (AB)-1 B-1A-1
Quite similar to (AB)T BTAT We will show the
identity (AB)-1 B-1A-1 later.
(4) For any positive integer n, An is
non-singular, and (An)-1 (A-1)n.
73
5.8 Properties of Inverses
If A and B are non-singular matrices of the same
order, and any scalar ? we have
(6) AT is non-singular and (AT)-1 (A-1)T.
(7) If AC 0, then C 0.
74
5.8 Properties of Inverses
If A and B are non-singular matrices of the same
order, and any scalar ? we have
Prove (AB)-1 B-1A-1
5E
75
P.164 Ex.5E
76
5.9 Some Illustrative Examples
77
5.9 Some Illustrative Examples
78
P.176 Ex.5F
79
Transformations of Points on the Coordinate Plane
80
5.10 Linear Transformations on the Rectangular
Cartesian Plane
81
5.10 Linear Transformations on the Rectangular
Cartesian Plane
82
5.10 Linear Transformations on the Rectangular
Cartesian Plane
83
5.10 Linear Transformations on the Rectangular
Cartesian Plane
84
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
85
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
86
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
87
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
88
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
89
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
90
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
91
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
92
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
93
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
(IV) Reflection
94
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
(IV) Reflection
The matrix representing the reflection about
95
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
96
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
97
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
98
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
99
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
100
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
101
5.11 Some special Linear Transformations on the
Rectangular Cartesian Plane
102
P.195 Ex.5G
