1 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
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CHAPTER 7
Matrices
I. Basic matrix operations
1. Matrix order
# rows x # Column
Example: A 3 x 4 matrix labelled A would be written a a a a
11 12 13 14
24
a
21
a
22
a
23
a
34
a
31
a
32
a
33
a
44
a
41
a
42
a
43
a
- # rows = 1; # columns = 0 row vector
# columns = 1; # rows = 0 column vector
2. Transposition
Matrix A (m x n) transposition A (n x m) found by replacing rows by columns
T
3. Addition and subtraction
- Condition: same order
-
Matrix A: a
11 12 13
a a
a
21
a
22
a
23
Matrix B: b b b
11 12 13
b
21
b
22
b
23
C = A + B = c c c where c = a + b ; c = a + b
11 12 13 11 11 11 12 12 12
c c c
21 22 23
4. Scalar multiplication
2 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
- Multiply a matrix A by a scalar k:
kA = ka11 ka12 ka13
ka21 ka22 ka23
5. Matrix multiplication.
- A (mA x nA) and B (mB x nB)
C = AB order (mA x nB)
Method:
SUMMARY
Provided that the indicated sums and products make sense,
A + B = B + A
A − A = 0
3 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
A + 0 = A
k (A + B) = kA + kB
k (lA) = ( kl ) A
A(B + C) = AB + AC
(A + B)C = AC + BC
A(BC) = (AB)C
We also have the non-property that, in general,
AB b BA
Explain: C = A x B condition: nA = mB
C = B x A condition: nB = mA
AB b BA
II. Matrix inversion
A
-1
A = I and AA = I
-1
Determinant:
= 0 singular
0 non singular inverse
Ax = b
by A gives
-1
A
-1
( Ax ) = A b
-1
(A
-1
A)x = A b (associative property)
-1
Ix = A b (definition of an inverse)
-1
x = A b
-1
Suy lu n t order c a 2 ma tr n. 2 ma tr n ch nhân ¿ ÿ ¿ ¿ ß
đ±ÿc vßi nhau s c ß ßt c a ma tr n 1 = s hàng c a ÿ ¿ ß ÿ
4 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
Matrix (2x2) a
11
a
12
a a
21 22
Matrix (3x3) a a
11
a
12 13
a a a
21 22 23
31
a a
32
a
33
Identity
matrices
I = 1 0
0 1
I = 1 0 0
0 1 0
0 0 1
Det (A)
= a a
11
a
22
12
a
21
Step 1: Cofactor (A )
ij
- The cofactor A = det(A) [matrix (2x2)] when
ij
delete row i and column j
- Prefixed by a 8–9 sign because from the pattern
Step 2: Det(A)
det(A) = a11A11 + a12A12 + a13A13
= a12A12 + a22A22 + a32A32
= a13A13 + a23A23 + a33A33
Inverse A
-
1
III. Cramer’s rule
Ax = b, where
A = a a a x = x b = b
11 12 13 1 1
a a a b
21 22 23
x
2 2
a
31 32
a a
33
x
3
b
3
Step 1: Tìm matrix A
i
Thay c t i = [b] Aß
i
Step 2: Cramers rule
i=1
2
3
5 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
6 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
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EXERCISE
M¿y b¿n nên gi i h t example c a t t c¿ ¿ ÿ ¿ ¿ i v i t lu n quan các bài luôn nha. Ch±¡ng này đß ß ¿
trng nh¿t là quen cách tính. Quen r i sß ¿ dß áp d ng. ÿ
Vßi tr¿c nghi link cách bßm, đây là ¿m máy, h¿u h¿t các ph¿n m¿y b u có th¿n đß ß gi¿i b¿ng máy
tính (Casio 570, 580, Vinacal áp d ) ÿng t±¡ng tự
BM MÁY: http://bitexedu.com/wp-content/uploads/2019/02/matrix-1.pdf
BÀI T P V N DNG
7 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
8 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
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X = (3B 2A) = 6 6
T
2 11
13 1
9 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
10 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ng c BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
Quiz Mr. Tri Anh (Sem 1 2020 2021) a) Chapter 7 b) Chapter 8

Preview text:

1 | M A T H F O R B U S I N E S S C H A P T E R 7 CHAPTER 7 Matrices
I. Basic matrix operations 1. Matrix order # rows x # Column
Example: A 3 x 4 matrix labelled A would be written a11 a12 a13 a14 a21 a22 a23 a24 a31 a32 a33 a34 a41 a42 a43 a44
- # rows = 1; # columns = 0 → row vector
# columns = 1; # rows = 0 → column vector
2. Transposition
Matrix A (m x n) → transposition AT (n x m) found by replacing r – ows by columns
3. Addition and subtraction
- Condition: same order - Matrix A: a11 a12 a13 a21 a22 a23 Matrix B: b11 b12 b13 b21 b22 b23  C = A + B = c11 c12 c13
where c11 = a11 + b11; c12 = a12 + b12 c21 c22 c23
4. Scalar multiplication
Ngô Minh Tuy¿t Ngọc – BABAIU19066
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2 | M A T H F O R B U S I N E S S C H A P T E R 7
- Multiply a matrix A by a scalar k: kA = ka11 ka12 ka13 ka21 ka22 ka23
5. Matrix multiplication.
- A (mA x nA) and B (mB x nB)
 C = AB → order (mA x nB) Method: SUMMARY
Provided that the indicated sums and products make sense, A + B = B + A A − A = 0
Ngô Minh Tuy¿t Ngọc – BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
3 | M A T H F O R B U S I N E S S C H A P T E R 7 A + 0 = A k (A + B) = kA + kB k (lA) = ( kl ) A A(B + C) = AB + AC
Suy lu¿n từ order cÿa 2 ma tr¿n. 2 ma tr¿n chß nhân (A + B)C = AC + BC
đ±ÿc vßi nhau  sß cßt cÿa ma tr¿n 1 = sß hàng cÿa A(BC) = (AB)C
We also have the non-property that, in general, AB b BA
Explain: C = A x B condition: nA = mB C = B x A condition: nB = mA ➔ AB b BA II. Matrix inversion A-1A = I and AA-1 = I Determinant: = 0 → singular
≠ 0 → non – singular → inverse Ax = b by A-1 gives A-1 ( Ax ) = A-1b
(A-1 A)x = A-1b (associative property) Ix = A-1b (definition of an inverse) x = A-1b
Ngô Minh Tuy¿t Ngọc – BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
4 | M A T H F O R B U S I N E S S C H A P T E R 7 Matrix (2x2) a11 a12
Matrix (3x3) a11 a12 a13 a21 a22 a21 a22 a23 a31 a32 a33 Identity I = 1 0 I = 1 0 0 matrices 0 1 0 1 0 0 0 1 Det (A)
= a11a22 a12a21
Step 1: Cofactor (Aij)
- The cofactor Aij = det(A) [matrix (2x2)] when delete row i and column j
- Prefixed by a 8–9 sign because from the pattern • Step 2: Det(A)
det(A) = a11A11 + a12A12 + a13A13 = a12A12 + a22A22 + a32A32 = a13A13 + a23A23 + a33A33 Inverse A- 1 III. Cramer’s rule Ax = b, where i=1 2 3 A = a11 a12 a13 x = x1 b = b1 a21 a22 a23 x2 b2 a31 a32 a33 x3 b3
Step 1: Tìm matrix Ai Thay cßt i = [b] → Ai
Step 2: Cramer’s rule
Ngô Minh Tuy¿t Ngọc – BABAIU19066
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5 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ngọc – BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
6 | M A T H F O R B U S I N E S S C H A P T E R 7 EXERCISE
M¿y b¿n nên gi¿i h¿t example cÿa t¿t c¿ các bài luôn nha. Ch±¡ng này đßi vßi t lu ự ¿n quan
trọng nh¿t là quen cách tính. Quen rßi s¿ dß áp dÿng. Vßi tr¿c nghi link cách b ßm, đây là
¿m máy, h¿u h¿t các ph¿n m¿y b¿n đßu có thß gi¿i b¿ng máy
tính (Casio 570, 580, Vinacal áp dÿng ) t±¡ng tự
BẤM MÁY: http://bitexedu.com/wp-content/uploads/2019/02/matrix-1.pdf
BÀI TP VN DNG
Ngô Minh Tuy¿t Ngọc – BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
7 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ngọc – BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
8 | M A T H F O R B U S I N E S S C H A P T E R 7 X = (3B – 2A)T = 6 6 2 11 13 1
Ngô Minh Tuy¿t Ngọc – BABAIU19066
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9 | M A T H F O R B U S I N E S S C H A P T E R 7
Ngô Minh Tuy¿t Ngọc – BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc
10 | M A T H F O R B U S I N E S S C H A P T E R 7
Quiz Mr. Tri Anh (Sem 1 2020 202 – 1) a) Chapter 7 b) Chapter 8
Ngô Minh Tuy¿t Ngọc – BABAIU19066
Contact: https://www.facebook.com/ngominhtuyetngoc