คณิตศาสตร์ ม.4 2564 - ครูสมาน

1. 2 4-6 2. power point 3. 1) power point 2) / 3) 4) 5 STEPs Learning to Question 1. power point 2. Learning to Search 1. Learning to Construct 2. x y 1. ax by cz d 2. z x, y ax by c a, b, c a,b,c, d 1. 3x 2y 5

2. x y xy 5 3. x 7 y 5z 11 4. 6x1 3x2 4x3 13 5. x2 2 y 1 6. 4x1 x2 3 7. x1 x22 10 8. x1 x2 3 3. 4. 5. a, b, c, d, e, f a, b c, d ax by e cx dy f xy xy 1 x 3y 9 2x y 8 x 3y 9 ---------------- 1 2x y 8 ---------------- 2 2 1 ; 2x 6y 18 ---------------- 3 3 2 ; 5y 10 y 10 5 y2 y2 1 x 32 9 x69 x 96 x3

x3 y2 2 2x 6y 4 3x 9 y 11 2x 6y 4 ---------------- 1 3x 9y 11 ---------------- 2 1 1; x 3y 2 ---------------- 3 2 3 3 ; 3x 9y 6 ---------------- 4 2 4; 05 xy 3 2x 6y 4 3x 9y 6 2x 6y 4 ---------------- 1 3x 9y 6 ---------------- 2 1 1; x 3y 2 ---------------- 3 2 3 1 ; 3x 9y 6 ---------------- 4 2 4; 00 xy x 3y 2 y t ,t x 2 3t yt Learning to Communicate 1. a, b, c, d, e, f a, b c, d ax by e cx dy f xy

2. 3. Learning to Service 1. 2. (K) // 60% (P) // 60% A 60 1 2 3 4 5 80 % 70 79 % 60 69 % 50 59 % 50 %

2 2564 1 1. () 2. 2.1 2.2 2.3 2.4 2.5 2.6 - 3. - 4. - - () 2



1 2 2 4 20 1. 2 1. 2. n x1, x2, ..., xn 3. K P A 1 2 3 4 5 1. 2. 3.

1. 2 4-6 2. power point 3. 1) / 2) / 5 STEPs Learning to Question 1. power point 2. Learning to Search Learning to Construct 1. 2. 1. 2. ax by e cx dy f xy 3. 4. 5.

n n2 a1, a2, ..., a n, b a1, a 2, ..., a n a1x1 + a2x2 + ... + a nxn = b n x1, x2, ..., xn x1, x2, ..., xn m m2 x1, x2, ..., xn n x1, x2, ..., xn a11x1 a12 x2 m b1 m2 a21x1 a22 x2 b2 m ... a1n xn ... a2n xn am1x1 am2 x2 ... amn xn bm n 1 x+ y+ z= 6 2x - 2y + 2z = 4 3x + 3y - 3z = 0 x + y + z = 6 ---------------- 1 2x - 2y + 2z = 4 ---------------- 2 3x + 3y - 3z = 0 ---------------- 3 12 x - y + z = 2 ---------------- 4 2 14 2y = 4 13 y= 2 3 x + y - z = 0 ---------------- 5 (4) + (5) 2x = 2 x= 1

(1) - (5) 2z = 6 z= 3 (x , y , z) = (1, 2, 3) 2 3x + y - z = 3 x + 2y + 4z = 2 4x + 3y + 3z = 7 3x + y - z = 3 ---------------- 1 ---------------- 2 x + 2y + 4z = 2 ---------------- 3 ---------------- 4 (1)+(2) ; 4x + 3y + 3z = 7 3-4 ; (4) 4x + 3y + 3z = 5 02 x,y,z (3) 3 x+ y= 2 x + 3y + z = 5 3x + y - z = 3 x + y = 2 ---------------- 1 x + 3y + z = 5 ---------------- 2 3x + y - z = 3 ---------------- 3 (2) + (3) 4x + 4y = 8 1 x y 2 ---------------- 4 4 00 (1) - (4) x,y z x y2 x t ,t y2x y 2t y 2t 2 3t 2 t z 3 3t 2 t z 3 z 2t 1

x t , y 2 t , z 2t 1 t Learning to Communicate 1. x1, x2, ..., xn m m2 x1, x2, ..., xn n x1, x2, ..., xn 2. 3. Learning to Service 1. 2. (K) // 60% // 60% 1 60 (P) A 1 2 3

4 5 80 % 70 79 % 60 69 % 50 59 % 50 %

2 2564 1 1. () 2. () 2.1 2 2.2 2.3 2.4 2.5 2.6 - 3. - 4. - -



3 4 26 2 1 3 m row n 1. mn m, n I 1. K 2. P 3. colum 1) 2) A 1 2 3 4 5 1. 2. 3.

1. 2 4-6 2. power point 3. 1) / 1) / 5 STEPs Learning to Question 1. power point 2. Learning to Search 1. 2. Learning to Construct 1. 2. 3.

4. 210725 5. 120424 111563 210725 003280 120424 111563 003280 mn m, n I m row n colum a11 a12 ... a1n 1 a 21 a 22 ... a 2n 2 a m1 a m2 ... a mn m 1 2 ... n 0 4 3 0 a ij (entry) i j ij i 1,2,3,..., m j 1,2,3,..., n mn mn mn (dimension of matrix) 2451 24 0382 0259 1510 3746

15 34 7 10 22 xy 3x 5y 31 4x z A a11 a12 a21 a21 a, b A aij m n A mn ij aij i 1,2,3,..., m j 1,2,3,..., n 2.10 =a24 0 2.11 =a34 2 3 4 15 2.12 =a33 6 2 A 8 10 12 0 7 11 6 2 2.1 =a13 -1 2.4 =a12 4 2.7 =a23 12 2.2 =a31 -7 2.5 =a22 -10 2.8 =a14 5 2.3 =a11 3 2.6 =a21 8 2.9 =a32 11 A A 22 A 34 A a11 a12 a11 a12 a13 a14 a21 a21 A a21 a22 a23 a24 a31 a32 a33 a34 A 53

a11 a12 a13 a21 a22 a23 A a31 a32 a33 a41 a42 a43 a51 a52 a53 A At ij A A = aij m n =A at m row ji n m ij aij At A= 3 5 At 3 8 8 4 54 1 23 1 40 At 2 9 4 A= 4 9 6 3 65 0 45 3 11 At 3 10 2 11 9 0 14 A= 1 9 00 2 14 Learning to Communicate 1. n colum mn m, n I A aij m n A mn i 1,2,3,..., m j 1,2,3,..., n A At ij A 2.

3. Learning to Service 1. 2. (K) // 60% 1) // 60% (P) 60 A 1 2 3 4 5 80 % 70 79 % 60 69 % 50 59 % 50 %

2 2564 1 1. () 2. () 2.1 2 2.2 2.3 2.4 2.5 2.6 - 3. - 4. -



1 4 4 4 26 1 1. K P 1. 2. A 3. 2 1) 1) 2) 1) 2) 3) 1. 2. 3. 1. 2 4-6

2. power point 3. 1) / m row n 1) / 5 STEPs 1. Learning to Question 2. power point 1. Learning to Search 2. Learning to Construct 1. 3. mn m, n I colum 4. 5. 6. 2 A aij m n b bij m n mn

AB aij bij i 1,2,3,..., m j 1,2,3,..., n A=B 1) A B 1 2) A B A 1 2 0 , B 32 4 55 1 2 3 2 01 2 21 a11 1, a12 2, a13 0 a21 1, a22 2, a23 3 b11 1, b12 2, b13 0 b21 1, b22 2, b23 3 AB 2 A 0 0 ,B 000 0 0 000 AB AB A 4 3 ,B 43 2 0 21 3 AB AB AB A ab 0 B ba 0 1 a2 b2 1 abab 4 AB AB AB 51 5 73 x3 1 y 2 2z 1 x35 ---------------- 1 y2 7 ---------------- 2 2z 1 3 ---------------- 3 1 x x 53 x2 2 y y 72

2z y5 3 6 2z 1 z2 2x y 4 x 3y 11 2x y 4 ---------------- 1 ---------------- 2 x 3y 11 ---------------- 3 2 (2) 2x 6 y 22 (3) - (1) 5y 18 y 18 y 18 5 5 (1) 2x 18 4 5 2x 4 18 5 x 1 5 x = - 3, y = 5 z= 2 Learning to Communicate 1. 2 2. 3. Learning to Service 1. 2.

(K) // 60% 1) // 60% 60 (P) 1) 2) (A) 1) 2) 3) 80 % 70 79 % 60 69 % 50 59 % 50 %

2 2564 1 1. () 2. () 2.1 2 2.2 2.3 2.4 2.5 2.6 3. - - 4. - -



1 5 5 4 26 1. 1 1. 2 2. 3. 2 K 1. P 2. A 1) 2) 3) 1) 2) 1) 2) 3) 1. 2. 3.

1. 2 4-6 2. power point 3. 1) / 1) / 5 STEPs Learning to Question 1. power point 2. Learning to Search 1. 2. Learning to Construct 1. . 2 2 1. 2. 2.

A aij B bij mn A B aij bij m n A B aij bij m n 7 AB AB A 1 5 3 ,B 37 1 AB AB 2 4 7 354 AB 13 57 31 23 45 74 2 12 4 5 1 11 AB 13 57 31 23 45 74 4 22 4 8 1 93 6 A 1 3 0 ,B 2 2 7 9 3 1 A BA 1 3 0 ,B 24 2 7 9 38 16 AB AB B A 8.1 2x 1 a2 74 93 6 0 15 75 2x 1 7 9 a2 4 3 a 24 3 2x 8 9 a5 2x 1 x 1 2 x 1 a5 2

8.1 a1 b 3 2a 3 82 2c 5d 4b a c 01 a1 3 8 b 2a 3 2 a48 b 2(4) 3 2 1 a4 b5 2 b3 2c 4b 0 5d ac 2c 4 3 0 5d 46 1 2c 12 0 5 d 10 1 2c 12 d 5 10 1 c6 b4 a 4,b 3,c 6 b 4 Learning to Communicate 1. 2. 3. Learning to Service 1. 2. (K) (P) // 1) 60% 2) 3) // 60% 1) 2) 60 (A) 1) 2)

3) 80 % 70 79 % 60 69 % 50 59 % 50 %

1 2564 2 1. 2. () 2.1 2.2 () 2.3 2 2.4 2.5 2.6 - 3. - 4. - - -



6 1 4 26 62 1. 1. 2. 3. mn 0m n 0 n I n I jk n n j k n jk 1 0 K 1) 2) 3) 4 P 1) 2) A 1) 2) 3)

1. 2. 3. 1. 2 4-6 2. power point 3. 1) / 1) / 5 STEPs Learning to Question 2) power point 3) Learning to Search 1. 2. Learning to Construct 1. 2. 2.

3. mA aij m n r rA raij m n 9 A 3 7 1 ,B 62 4 06 4 2 5 15 9 C 81 34 2A 2A 2 3 7 1 6 14 2 4 2 5 8 4 10 3B 3B 3 6 2 4 18 6 12 1 5 9 3 15 27 9.3 3A 2B 3A 2B 3 3 7 1 2 6 2 4 4 2 5 1 5 9 9 21 3 12 4 8 12 6 15 2 10 18 9 12 21 4 38 12 2 6 10 15 18 3 25 11 14 16 3 mn 0 0m n 0 02 3 03 1 3 000 000 0 0 0

0000 01 3 n I n I jk n n j k n I jk 1, jk 0, jk In identity matrix n n In I 1 I1 1 I2 10 01 100 I3 0 1 0 001 Learning to Communicate 1 2 3 Learning to Service 1. 2.

(K) // 1) 60% 2) 3) // 4 60% (P) 60 1) 2) (A) 1) 2) 3) 80 % 70 79 % 60 69 % 50 59 % 50 %

2 1 - 2564 1. () 2. - 2.1 2.2 2.3 2.4 2.5 2.6 - 3. - - 4. - - - () 2



7 1 4 26 72 1. 1. 2. 3. inverse of a matrix A AB BA In B 1 A1 22 A1 B ad bc db ca K 1) 2) P 1) 2) A 1) 2) 3) 1. 2. 3.

1. 2 4-6 2. power point 3. 1) / 1) / 5 STEPs Learning to Question 1. power point 2. Learning to Search 1. 2. Learning to Construct 1. . aa 1 a 1a 1 a a1 . . .

A nn B nn AB BA I n B inverse of a matrix A B A1 1 A 21 B 21 32 32 AB BA I n AB 21 2 1 32 3 2 43 22 10 66 34 01 BA 2 12 1 3 2 32 43 22 10 01 66 34 B A1 A A 12 34 2 A1 ab cd AA 1 1 2 a b 34c d a 2c b 2d 3a 4c 3b 4d AA 1 I n 10 a 2c b 2d 01 3a 4c 3b 4d a 2c 1 1 2 3a 4c 0 3 4 b 2d 0 5 3b 4d 1 2 1 ; 2a 4c 2 2 5; a2

a2 1 2 2c 1 2c 3 c 3 2 2 3; 2b 4d 0 6 4 6; b1 b1 3 1 2d 0 d 1 2 21 A1 3 1 22 22 A ab ad bc 0 cd A1 1 d b ad bc c a 3 A 11 A1 10 A1 1 01 4 10 11 1 1 A1 10 1 01 1 1 10 1 11 1 01 11 A 01 A1 23 1 31 03 21 2 0 13 1 02 2 0

13 1 220 31 22 10 5 A 42 A1 13 A1 1 3 2 43 21 1 4 13 2 12 2 1 4 13 2 10 1 4 32 10 10 14 10 10 6 A 13 A1 26 A1 1 63 1 6 32 2 1 16 3 02 1 ad bc = 0 A 7 A 46 A1 23 A1 36 1 24 ad bc = 0 A 43 6 2 13 6 02 4

A nonsingular matrix 1. A singular matrix A 2. A A 5 A x x4 x A 2x ad bc 0 x2 2(x 4) 0 x2 2x 8 0 (x 4)(x 2) 0 x 2, 4 x 2, 4 Learning to Communicate 1. inverse of a matrix AB BA I n B 22 AB A1 A1 1 d b ad bc c a 2. 3. Learning to Service 1. 2.