Transcript Document
a11 a12
a 21 a 22
a
31 a 32
a13
a 23
a 33
b11 b12
b 21 b 22
b
31 b 32
b13
b 23
b 33
=
Matrix
Multiplication
a11b11 a12b21 a13b31
a11b12 a12b22 a13b33
a11b13 a12b23 a13b33
a 21b11 a 22b21 a 23b31
a 21b12 a 22b22 a 23b33
a 21b13 a 22b23 a 23b33
a 31b11 a 32b21 a 33b31
a 31b12 a 32b22 a 33b33
a 31b13 a 32b23 a 33b33
Cofactor
Aij
a11
a12
a13
a 21 a 22
a 23
a 31
a 33
a 32
of the (i,j)th entry
(-1)i+j (determinant obtained by deleting the
ith row and jth column of the matrix)
AA
1132
1a1 a 22 a a 23
( 1) 11
a 32 13a 33
a 21 a 33
1 2
a 21 a 23
A12 (1) a a
a11 31 a13 33
A22 a a
31
33
13 a 21 a 22
A13 (1) a a
31
32
The sign of (-1)i+j is given by
next
a11
a12
a 21 a 22
a 31 a 32
a13
a 23
a 33
Determinant
+a11a22a33 +a21a32a13 +a31a12a23
– a31a22a13 – a21a12a33 – a11a32a23
= a11 A11+ a12 A12+ a13A13
a 22
a11
a 32
a 21
a 23
a
– 12
a 31
a 33
direct
expansion
expanded
along
a13
a 21 a 22 st
a
1 row
a 33 + 13 a 31 a 32
= a21 A21+ a22 A22+ a23A23 a 21 a12 a13 a 22 a11 a13 a 23 a11 a12 2nd row
a 32 a 33
a12 a13
a 31 a 33
a11 a13
a 31 a 32
a11 a12
= a31 A31+ a32 A32+ a33A33 a 31
a(-1)
a 33by
32 i+j is given
The sign of
a 22 a 23
= a11 A11+ a21 A21+ a31A31
= a12 A12+ a22 A22+ a32A32
= a13 A13+ a23 A23+ a33A33
3rd row
a 21 a 23
a 21 a 22
a a a a
first
a 22 a 23
21
23
21
22
a11
a12
a13
column
a 32 a 33 a 31 a 33 a 31 a 32
a 21 a 23
a11
a13
a11 a13 second
a12
a 22
a 32
a 31 a 33
a 31 a 33
a 21 a 23 column
a12 a13
a11 a12
a11 a12
third
a 31
a 23
a 33
a 22 a 23
a 31 a 32
a 21 a 22
column
A
1
Inverse
1
(A ij ) t
det A
Proof
a11
1
1
t
A
(A ij )
a 21
det A
det A
a 31
a11
1
a 21
det A
a 31
a12
a 22
a 32
a12
a 22
a 32
a13 A11
a 23 A12
a 33 A13
a13 A11
a 23 A 21
a 33 A 31
A 21
A 22
A 23
A12
A 22
A 32
A13
A 23
A 33
t
A 31
A 32
A 33
a11A11 adetA
A22+a13A23
12 A12 a13A13 a11A21+a120
0
1
A12+a23A13 a 21A 21 adetA
a21A11+a220
22 A 22 a 23A 23
0
det A
a
A
a
A
a
A
31 31 detA
32 32
33 33
0
0
=I.
Explanation
Similarly, it can be proved that
Explanation
1
( A ij ) t A I. A-1 hence found.
det A
END
a11
a12
a 21 a 22
a 31 a 32
a13
a 23
a 33
Proof
+a11a22a33 +a21a32a13 +a31a12a23
– a31a22a13 – a21a12a33 – a11a32a23
=a11(a22a33-a32a23) – a12(a21a33-a31a13)+ a13(a21a32-a31a22)
a 21 a13a 21 aa2321 a 22
a 22 a 23
a 21
a11
a
– a12 – a12 + a13
a32 a33
a 31 a 33a 31 aa3331 a 32+ 13 a 31
= a11 A11 + a12 A12 + a13A13
a 22
a 32
(expanded along first row)
The sign of (-1)i+j is given by
back
a11
a12
a13
a 21 a 22
a 23
a 31
a 33
a 32
a11What
A11+about
a12 A12+ a13A13 = det A
a11 A21+ a12 A22+ a13A23?
a21 A21+ a22 A22+ a23A23 = det A
a31 A31+ a32 A32+ a33A33 = det A
Note that
ax21
A + ay12A22 + a13
z A23=
23
11 21 a22
Proof2
0
Expand along first row
Expand along second row
Expand along third row
a11 a12 a13
aax21
aya22
az 23
11
12 a13
a 31 a 32 a 33
back to
inverse
along 2nd row
a21 A11+ a22 A12+ a23A13?
along 1st row
a 21 a 22
a 23
a 21 a 22
a 31 a 33
a 23
a 33
=0