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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Help sessions:
The Math 304/309/311/323 Help Session will be
held on MW from 7:3010:00pm and on TR from
6:309:00 p.m., BLOC 160
Homework assignment #3
(due Thursday, September 22)
All problems are from Leon’s book (8th edition). Section 2.1: 3b, 3c, 3d, 3e, 3f, 3g
Section 2.2: 3f, 4, 7c, 7d MATH 304
Linear Algebra
Lecture 6:
Transpose of a matrix.
Determinants. Transpose of a matrix
Deﬁnition. Given a matrix A, the transpose of A,
denoted AT , is the matrix whose rows are columns
of A (and whose columns are rows of A). That is,
if A = (aij ) then AT = (bij ), where bij = aji . 14
T
123
Examples.
= 2 5 ,
456
36 T
7
T
47
47
8 = (7, 8, 9),
=
.
70
70
9 Properties of transposes:
• (AT )T = A
• (A + B )T = AT + B T
• (rA)T = rAT
• (AB )T = B T AT
• (A1 A2 . . . Ak )T = AT . . . AT AT
21
k
• (A−1)T = (AT )−1 Deﬁnition. A square matrix A is said to be
symmetric if AT = A.
For example, any diagonal matrix is symmetric.
Proposition For any square matrix A the matrices
B = AAT and C = A + AT are symmetric.
Proof:
B T = (AAT )T = (AT )T AT = AAT = B ,
C T = (A + AT )T = AT + (AT )T = AT + A = C . Determinants
Determinant is a scalar assigned to each square matrix.
Notation. The determinant of a matrix
A = (aij )1≤i ,j ≤n is denoted det A or
a11 a12
a21 a22
.
.
.
.
.
.
an1 an2 . . . a1n
. . . a2n
... . .
.
.
. . . ann Principal property: det A = 0 if and only if a
system of linear equations with the coeﬃcient
matrix A has a unique solution. Equivalently,
det A = 0 if and only if the matrix A is invertible. Deﬁnition in low dimensions
Deﬁnition. det (a) = a, ab
= ad − bc ,
cd a11 a12 a13
a21 a22 a23 = a11a22 a33 + a12a23a31 + a13a21a32 −
a31 a32 a33
−a13a22a31 − a12 a21a33 − a11a23 a32. +: −: * ∗
∗ ∗ ∗
* ∗
*
∗
∗
*
∗ ∗
∗ ∗ , ∗
*
* *
∗ ∗ , *
∗
∗ *
∗
∗
*
∗
∗ ∗ * , ∗ ∗ ∗ , * ∗
*
∗ ∗
∗
* *
∗
∗ ∗
∗
* * ∗ .
∗ ∗ * .
∗ Examples: 2×2 matrices
10
= 1,
01
−2 5
= − 6,
03 30
= − 12,
0 −4
70
= 14,
52 0 −1
= 1,
10 00
= 0,
41 −1 3
= 0,
−1 3 21
= 0.
84 Examples: 3×3 matrices
3 −2 0
1 0 1 = 3 · 0 · 0 + (−2) · 1 · (−2) + 0 · 1 · 3 −
−2 3 0
− 0 · 0 · (−2) − (−2) · 1 · 0 − 3 · 1 · 3 = 4 − 9 = −5,
146
0 2 5 =1·2·3+4·5·0+6·0·0−
003
− 6 · 2 · 0 − 4 · 0 · 3 − 1 · 5 · 0 = 1 · 2 · 3 = 6. General deﬁnition
The general deﬁnition of the determinant is quite
complicated as there is no simple explicit formula.
There are several approaches to deﬁning determinants.
Approach 1 (original): an explicit (but very
complicated) formula.
Approach 2 (axiomatic): we formulate
properties that the determinant should have.
Approach 3 (inductive): the determinant of an
n×n matrix is deﬁned in terms of determinants of
certain (n − 1)×(n − 1) matrices. Mn (R): the set of n×n matrices with real entries.
Theorem There exists a unique function
det : Mn (R) → R (called the determinant) with the
following properties:
• if a row of a matrix is multiplied by a scalar r ,
the determinant is also multiplied by r ;
• if we add a row of a matrix multiplied by a scalar
to another row, the determinant remains the same;
• if we interchange two rows of a matrix, the
determinant changes its sign;
• det I = 1. Corollary 1 Suppose A is a square matrix and B is
obtained from A applying elementary row operations.
Then det A = 0 if and only if det B = 0.
Corollary 2 det B = 0 whenever the matrix B has
a zero row.
Hint: Multiply the zero row by the zero scalar. Corollary 3 det A = 0 if and only if the matrix A
is not invertible.
Idea of the proof: Let B be the reduced row echelon form of
A. If A is invertible then B = I ; otherwise B has a zero row. Row echelon form of a square matrix A: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ det A = 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ det A = 0 3 −2 0
Example. A = 1 0 1, det A =?
−2 3 0
In the previous lecture we have transformed the
matrix A into the identity matrix using elementary
row operations:
•
•
•
•
•
•
•
• interchange the 1st row with the 2nd row,
add −3 times the 1st row to the 2nd row,
add 2 times the 1st row to the 3rd row,
multiply the 2nd row by −0.5,
add −3 times the 2nd row to the 3rd row,
multiply the 3rd row by −0.4,
add −1.5 times the 3rd row to the 2nd row,
add −1 times the 3rd row to the 1st row. 3 −2 0
Example. A = 1 0 1, det A =?
−2 3 0
In the previous lecture we have transformed the
matrix A into the identity matrix using elementary
row operations.
These included two row multiplications, by −0.5
and by −0.4, and one row exchange.
It follows that
det I = − (−0.5) (−0.4) det A = (−0.2) det A.
Hence det A = −5 det I = −5. Other properties of determinants
• If a matrix A has two
det A = 0.
a1 a2
b1 b2
a1 a2 identical rows then
a3
b3 = 0
a3 • If a matrix A has two rows proportional then
det A = 0.
a1 a2 a3
a1 a2 a3
b1 b2 b3 = r b1 b2 b3 = 0
a1 a2 a3
ra1 ra2 ra3 Distributive law for rows
• Suppose that matrices X , Y , Z are identical
except for the i th row and the i th row of Z is the
sum of the i th rows of X and Y .
Then det Z = det X + det Y .
′
′
′
′
′
′
a1 +a1 a2+a2 a3 +a3
a1 a2 a3
a1 a2 a3
b1
b2
b3
= b1 b2 b3 + b1 b2 b3
c1
c2
c3
c1 c2 c3
c1 c2 c3 • Adding a scalar multiple of one row to another
row does not change the determinant of a matrix.
a1 + rb1 a2 + rb2 a3 + rb3
=
b1
b2
b3
c1
c2
c3
a1 a2 a3
rb1 rb2 rb3
a1 a2 a3
= b1 b2 b3 + b1 b2 b3 = b1 b2 b3
c1 c2 c3
c1 c2 c3
c1 c2 c3 Deﬁnition. A square matrix A = (aij ) is called
upper triangular if all entries below the main
diagonal are zeros: aij = 0 whenever i > j .
• The determinant of an upper triangular matrix is
equal to the product of its diagonal entries.
a11 a12 a13
0 a22 a23 = a11a22a33
0 0 a33
• If A = diag(d1, d2, . . . , dn ) then
det A = d1 d2 . . . dn . In particular, det I = 1. Determinant of the transpose • If A is a square matrix then det AT = det A.
a1 b1 c1
a1 a2 a3
a2 b2 c2 = b1 b2 b3
a3 b3 c3
c1 c2 c3 Columns vs. rows
• If one column of a matrix is multiplied by a
scalar, the determinant is multiplied by the same
scalar.
• Interchanging two columns of a matrix changes
the sign of its determinant.
• If a matrix A has two columns proportional then
det A = 0.
• Adding a scalar multiple of one column to
another does not change the determinant of a
matrix. Submatrices
Deﬁnition. Given a matrix A, a k ×k submatrix
of A is a matrix obtained by specifying k columns
and k rows of A and deleting the other columns and
rows. 1234
∗2∗4
10 20 30 40 → ∗ ∗ ∗ ∗ → 2 4
59
3579
∗5∗9 Given an n×n matrix A, let Mij denote the
(n − 1)×(n − 1) submatrix obtained by deleting the
i th row and the j th column of A. 3 −2 0
Example. A = 1 0 1.
−2 3 0
M11 =
M21 =
M31 = 01
, M12 =
30 11
, M13 =
−2 0 10
,
−2 3 −2 0
, M22 =
30 30
, M23 =
−2 0 3 −2
,
−2 3 −2 0
, M32 =
01 30
, M33 =
11 3 −2
.
10 Row and column expansions
Given an n×n matrix A = (aij ), let Mij denote the
(n − 1)×(n − 1) submatrix obtained by deleting the
i th row and the j th column of A.
Theorem For any 1 ≤ k , m ≤ n we have that
n (−1)k +j akj det Mkj , det A =
j =1 (expansion by kth row )
n (−1)i +maim det Mim . det A =
i =1 (expansion by mth column) Signs for row/column expansions +
− + −
.
.
. −
+
−
+
.
.
. +
−
+
−
.
.
. −
+
−
+
.
.
. ···
· · · · · · · · ·
... 123
Example. A = 4 5 6.
789
Expansion by the 1st row: ∗2∗
∗∗ 3
1 ∗∗ ∗ 5 6 4 ∗ 6 4 5 ∗ ∗ 89
7∗9
78 ∗
det A = 1 56
46
45
−2
+3
89
79
78 = (5 · 9 − 6 · 8) − 2(4 · 9 − 6 · 7) + 3(4 · 8 − 5 · 7) = 0. 123
Example. A = 4 5 6.
789
Expansion by the 2nd 1∗
∗2∗
4 ∗ 6 ∗ 5
7∗9
7∗
det A = −2 column: 3
1∗3
∗ 4 ∗ 6 9
∗8∗ 46
13
13
+5
−8
79
79
46 = −2(4 · 9 − 6 · 7) + 5(1 · 9 − 3 · 7) − 8(1 · 6 − 3 · 4) = 0. 123
Example. A = 4 5 6.
789
Subtract the 1st row from the 2nd row and from
the 3rd row:
123
123
123
4 5 6 = 3 3 3 = 3 3 3 =0
789
789
666
since the last matrix has two proportional rows. ...
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 Spring '08
 HOBBS
 Linear Algebra, Algebra

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