chapter 2 section 2. lemma 2.2.1 let i=1 and j=2, then
TRANSCRIPT
Chapter 2
Section 2
Lemma 2.2.1
€
A =
1 2 3
3 4 2
2 1 5
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Let i=1 and j=2, then
€
ai1A j1 + ai2A j 2 + ai3A j 3= a11A21 + a12A22 + a13A23
= A21 + 2A22 + 3A23
= −2 3
1 5+ 2
1 3
2 5− 3
1 2
2 1
= −(10 − 3) + 2(5 − 6) − 3(1− 4)
= −7 − 2 + 9 = 0
Lemma 2.2.1
€
A =
1 2 3
3 4 2
2 1 5
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Let i=1 and j=2, then
€
ai1A j1 + ai2A j 2 + ai3A j 3= a11A21 + a12A22 + a13A23
= A21 + 2A22 + 3A23
= −2 3
1 5+ 2
1 3
2 5− 3
1 2
2 1
= −(10 − 3) + 2(5 − 6) − 3(1− 4)
= −7 − 2 + 9 = 0
Lemma 2.2.1
€
A =
1 2 3
3 4 2
2 1 5
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Let i=1 and j=2, then
€
ai1A j1 + ai2A j 2 + ai3A j 3= a11A21 + a12A22 + a13A23
= A21 + 2A22 + 3A23
= −2 3
1 5+ 2
1 3
2 5− 3
1 2
2 1
= −(10 − 3) + 2(5 − 6) − 3(1− 4)
= −7 − 2 + 9 = 0
Lemma 2.2.1
€
A =
1 2 3
3 4 2
2 1 5
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Let i=1 and j=2, then
€
ai1A j1 + ai2A j 2 + ai3A j 3= a11A21 + a12A22 + a13A23
= A21 + 2A22 + 3A23
= −2 3
1 5+ 2
1 3
2 5− 3
1 2
2 1
= −(10 − 3) + 2(5 − 6) − 3(1− 4)
= 7 + 2 − 9 = 0
Lemma 2.2.1
€
A =
1 2 3
3 4 2
2 1 5
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Let i=1 and j=2, then
€
ai1A j1 + ai2A j 2 + ai3A j 3= a11A21 + a12A22 + a13A23
= A21 + 2A22 + 3A23
= −2 3
1 5+ 2
1 3
2 5− 3
1 2
2 1
= −(10 − 3) + 2(5 − 6) − 3(1− 4)
= 7 + 2 − 9 = 0
Lemma 2.2.1
€
A =
1 2 3
3 4 2
2 1 5
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Let i=1 and j=2, then
€
ai1A j1 + ai2A j 2 + ai3A j 3= a11A21 + a12A22 + a13A23
= A21 + 2A22 + 3A23
= −2 3
1 5+ 2
1 3
2 5− 3
1 2
2 1
= −(10 − 3) + 2(5 − 6) − 3(1− 4)
= −7 − 2 + 9 = 0
Lemma 2.2.1Notice
€
=−2 3
1 5+ 2
1 3
2 5− 3
1 2
2 1
= det(B)
where
€
B =
1 2 3
1 2 3
2 1 5
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
€
A =
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Switching Row 2 and Row 3, and calculating the determinant,
€
a11 a12 a13
a31 a32 a33
a21 a22 a23
= a11
a32 a33
a22 a23
− a12
a31 a33
a21 a23
+ a13
a31 a32
a21 a22
€
=−a11
a22 a23
a32 a33
+ a12
a21 a23
a31 a33
− a13
a21 a22
a31 a32
€
=− a11
a22 a23
a32 a33
− a12
a21 a23
a31 a33
+ a13
a21 a22
a31 a32
⎛
⎝ ⎜
⎞
⎠ ⎟
€
=−det(A)
€
A =
a11 a12 a13
a21 a22 a23
a31 a32 a33
⎡
⎣
⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥
Multiplying Row 3 by 4 and calculating the determinant,
€
a11 a12 a13
a21 a22 a23
4a31 4a32 4a33
= a11
a22 a23
4a32 4a33
− a12
a21 a23
4a31 4a33
+ a13
a21 a22
4a31 4a32
€
=4a11
a22 a23
a32 a33
− 4a12
a21 a23
a31 a33
+ 4a13
a21 a22
a31 a32
€
=4 a11
a22 a23
a32 a33
− a12
a21 a23
a31 a33
+ a13
a21 a22
a31 a32
⎛
⎝ ⎜
⎞
⎠ ⎟
€
=4 det(A)
• If E is an elementary matrix, then
where
€
det(EA) = det(E)det(A)
€
det(E) =
−1
α ≠ 0
1
⎧
⎨ ⎪
⎩ ⎪
If E is of type I
If E is of type II
If E is of type III
• Similar results hold for column operations. Indeed, if E is an elementary matrix, then ET is also an elementary matrix and
€
det(AE)
= det((AE)T )
= det(E TAT )
= det(E T )det(AT )
= det(E)det(A)
• Similar results hold for column operations. Indeed, if E is an elementary matrix, then ET is also an elementary matrix and
€
det(AE)
= det((AE)T )
= det(E TAT )
= det(E T )det(AT )
= det(E)det(A)
• Similar results hold for column operations. Indeed, if E is an elementary matrix, then ET is also an elementary matrix and
€
det(AE)
= det((AE)T )
= det(E TAT )
= det(E T )det(AT )
= det(E)det(A)
• Similar results hold for column operations. Indeed, if E is an elementary matrix, then ET is also an elementary matrix and
€
det(AE)
= det((AE)T )
= det(E TAT )
= det(E T )det(AT )
= det(E)det(A)
I. Interchanging two rows (or columns) of a matrix changes the sign of the determinant.
II. Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar.
III. Adding a multiple of one row (or column) to another does not change the value of the determinant
I. Interchanging two rows (or columns) of a matrix changes the sign of the determinant.
II. Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar.
III. Adding a multiple of one row (or column) to another does not change the value of the determinant
I. Interchanging two rows (or columns) of a matrix changes the sign of the determinant.
II. Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar.
III. Adding a multiple of one row (or column) to another does not change the value of the determinant
I. Interchanging two rows (or columns) of a matrix changes the sign of the determinant.
II. Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar.
III. Adding a multiple of one row (or column) to another does not change the value of the determinant
I. Interchanging two rows (or columns) of a matrix changes the sign of the determinant.
II. Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar.
III. Adding a multiple of one row (or column) to another does not change the value of the determinant
I. Interchanging two rows (or columns) of a matrix changes the sign of the determinant.
II. Multiplying a single row (or column) of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar.
III. Adding a multiple of one row (or column) to another does not change the value of the determinant