Soal Matriks dan Ruang Vektor
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Transcript of Soal Matriks dan Ruang Vektor
1 | P a g e
SOAL NO.1
A = B = C =
a. Adj (A) + K12 Adj(B)
Jawab :
A11
= (-1)1+1
= + (-1 - 4) = + (-5) = -5
A12
= (-1)1+2
= - (0 - 10) = - (-10) = 10
A13
= (-1)1+3
= + (0 + 5) = + (5) = 5
A21
= (-1)2+1
= - (4 - 4) = - (0) = 0
A22
= (-1)2+2
= + (1 - 10) = + (-9) = -9
A23
= (-1)2+3
= - (2 - 20) = - (-18) = 18
A31
= (-1)3+1
= + (8 + 2) = + (10) = 10
A32
= (-1)3+2
= - ( 2 - 0) = - (2) = -2
A33
= (-1)3+3
= + (-1 - 0) = +(-1) = -1
Sehingga ,
Adj (A) =
=
B11
= (-1)1+1
= + (-5 - 3) = + (-8) = -8
B12
= (-1)1+2
= - (4 - 3) = - (1) = -1
B13
= (-1)1+3
= + (4 + 5) = + (9) = 9
B21
= (-1)2+1
= - (4 + 2) = - (6) = -6
B22
= (-1)2+2
= + (8 + 2) = + (10) = 10
Sehingga ,
2 | P a g e
Adj(B) = K12 Adj(B) =
Adj (A) + K12 Adj(B)
= + =
B23
= (-1)2+3
= - (8 - 4) = - (4) = -4
B31
= (-1)3+1
= + (12 - 10) = + (2) = 2
B32
= (-1)3+2
= - ( 24 + 8) = - (32) = -32
B33
= (-1)3+3
= + (-40 – 16) = +(- 56) = -56
Adj (B) =
=
3 | P a g e
b. H31(-2)
(C -1
) – Adj (B) . Adj (A)
Det C= = (-80 + 36 – 24 – 36 – 16 – 120 ) = -240
C-1
=
= : -240
=
C11
= (-1)1+1
= + (-16 - 24) = + (- 40) = -40
C12
= (-1)1+2
= - (16 - 36) = - (-20) = 20
C13
= (-1)1+3
= + (8 + 12) = + (20) = 20
C21
= (-1)2+1
= - (8 + 12) = - (20) = -20
C22
= (-1)2+2
= + (40 + 18) = + (58) = 58
C23
= (-1)2+3
= - (- 20 – 6) = - (14) = -14
C31
= (-1)3+1
= + (6 - 6) = + (0) = 0
C32
= (-1)3+2
= - ( 30 + 6) = - (36) = -36
C33
= (-1)3+3
= + (-10 - 2) = +(-12) = -12
Sehingga ,
Adj (C) =
=
4 | P a g e
Adj (B) x Adj (A)
= x =
H31(-2)
(C -1
) = H31(-2)
H3 =
H1 . (-2) =
______________+
H31(-2)
(C -1
)
H31(-2)
(C -1
) – Adj (B) . Adj (A)
= - =
5 | P a g e
c. Adj[ H12(-1)
(A)] + A-1
B-1
Det (A) = = -1 + 40 +0 +10 – 0 – 4 = 45
A-1
= = : 45 =
Det (B) = = - 40 + 12 – 8 – 10 – 16 – 24 = - 86
B-1
= = : -86 =
A-1
B-1 = x =
6 | P a g e
H12(-1)
(A) = H12(-1)
= H1 = 1 4 2 H12(-1)
(A)
H2 . (-1) = 0 1 -2 + 1 5 0
Misal H12(-1)
(A) = X Maka :
X11
= (-1)1+1
= + (-1 - 4) = + (-5) = -5
X12
= (-1)1+2
= - (0 - 10) = - (-10) = 10
X13
= (-1)1+3
= + (0 + 5) = + (5) = 5
X21
= (-1)2+1
= - (5 - 0) = - (5) = -5
X22
= (-1)2+2
= + (1 - 0) = + (1) = 1
X23
= (-1)2+3
= - (2 - 25) = - (-23) = 23
X31
= (-1)3+1
= + (10 - 0) = + (10) = 10
X32
= (-1)3+2
= - ( 2 - 0) = - (2) = -2
X33
= (-1)3+3
= + (-1 - 0) = +(-1) = -1
Sehingga ,
Adj (X) =
=
Adj[ H12(-1)
(A)] + A-1
B-1
= + =
7 | P a g e
d. (AB)-1
+ Adj (AB)
AB = x =
= = -549 + 686 – 264 – 4116 – (-286) – (-84) = -3870
AB11
= (-1)1+1
= + (-21 – (-11)) = + (-10) = -10
AB12
= (-1)1+2
= - (6 – (-49)) = - (55) = -55
AB13
= (-1)1+3
= + (-22 - 343) = + (- 365) = -365
AB21
= (-1)2+1
= - (42 - 132) = - (-90) = 90
AB22
= (-1)2+2
= + (-78 - 588) = + (-666) = -666
AB23
= (-1)2+3
= - (286 – (-686)) = - (972) = -972
AB31
= (-1)3+1
= + (14 - 84) = + (-70) = -70
AB32
= (-1)3+2
= - ( -26 – (-24)) = - (-2) = 2
AB33
= (-1)3+3
= + (182 - 28) = +(154) = 154
Sehingga ,
Adj (AB)
=
=
AB-1
= = : -3870 =
8 | P a g e
(AB)-1
+ Adj (AB)
= +
=
9 | P a g e
e. [Adj (CA)-1
] + Adj(A)
CA = x =
CA11
= (-1)1+1
= + (616 – 216) = + (400) = -400
CA12
= (-1)1+2
= - (896 - 276) = - (620) = -620
CA13
= (-1)1+3
= + (1152 - 1012) = + ( 140 ) = -140
CA21
= (-1)2+1
= - (364 - 324) = - (40) = - 40
CA22
= (-1)2+2
= + (-280 - 414) = + (-694) = -694
CA23
= (-1)2+3
= - (-360 - 598) = - (- 958) = 958
CA31
= (-1)3+1
= + (78 - 198) = + (-120) = -120
CA32
= (-1)3+2
= - ( -60 - 288) = - (-348) = 348
CA33
= (-1)3+3
= + (-220 - 416) = +(-636) = -636
Sehingga ,
Adj (AB)
=
=
Misal Adj (CA) = X , Maka :
Det (X) = = 176553600-1948800+71275200-11659200+15772800-
133353600 = 116.640.000
X11
= (-1)1+1
= + (441384 - 333384) = + (108000) = 108000
X12
= (-1)1+2
= - (394320 - 48720) = - (345600) = -345600
X13
= (-1)1+3
= + (-593960 – (-97160)) = + ( -496800 ) = -496800
X21
= (-1)2+1
= - (25440-(-114960)) = - (140400) = - 140400
X22
= (-1)2+2
= + (-254400 – (-16800)) = + (-237600) = -237600
10 | P a g e
[Adj (CA)-1
] + Adj(A)
= +
=
X23
= (-1)2+3
= - (383200-(-5600)) = - (388800) = -388800
X31
= (-1)3+1
= + ( - 13920 - 83280) = + (-97200) = -97200
X32
= (-1)3+2
= - ( 139200 - 74400) = - (64800) = -64800
X33
= (-1)3+3
= + (-277600 - 24800) = +(-302400) = -302400
Sehingga ,
Adj (X)
= =
X-1
= = : 116640000
=