Exercise Zone : Matriks [2]

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Tipe :

Standar SBMPTN

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No. 11

Diketahui persamaan matriks $\left(\begin{array}{cc}3x& 3y\\ 6& 18\end{array}\right)-2\left(\begin{array}{cc}x& 6\\ -1& y+1\end{array}\right)=\left(\begin{array}{cc}2& 2x-y\\ 8& 8\end{array}\right)$. Nilai dari {x-y=}

1. -2
2. 0
3. 2

4. 4
5. 6

$\begin{array}{rl}\left(\begin{array}{cc}3x& 3y\\ 6& 18\end{array}\right)-2\left(\begin{array}{cc}x& 6\\ -1& y+1\end{array}\right)& =\left(\begin{array}{cc}2& 2x-y\\ 8& 8\end{array}\right)\\ \left(\begin{array}{cc}3x& 3y\\ 6& 18\end{array}\right)-\left(\begin{array}{cc}2x& 12\\ -2& 2y+2\end{array}\right)& =\left(\begin{array}{cc}2& 2x-y\\ 8& 8\end{array}\right)\\ \left(\begin{array}{cc}x& 3y-12\\ 8& -2y+16\end{array}\right)& =\left(\begin{array}{cc}2& 2x-y\\ 8& 8\end{array}\right)\end{array}$
x=\boxed{2}
$\begin{array}{rl}-2y+16& =8\\ -2y& =-8\\ y& =\boxed{{\displaystyle 4}}\end{array}$

$\begin{array}{rl}x-y& =2-4\\ & =\boxed{{\displaystyle \boxed{{\displaystyle -2}}}}\end{array}$

Jadi, x-y=-2.
JAWAB: A

No. 12

Diberikan matriks $P=\left(\begin{array}{cc}3& -1\\ 5& 2\end{array}\right)$ dan $Q=\left(\begin{array}{cc}3r& 2\\ r& p+1\end{array}\right)$ dengan {r\neq0} dan {p\neq0}. Supaya matriks PQ tidak mempunyai invers, maka nilai {3p + 2 =}

1. 4
2. 3
3. 2

4. 1
5. 0

Tidak punya invers artinya det = 0.
$\begin{array}{rl}\left|PQ\right|& =0\\ |P||Q|& =0\end{array}$
Karena |P|\neq0 maka
$\begin{array}{rl}|Q|& =0\\ 3r(p+1)-2r& =0\\ 3pr+3r-2r& =0\\ 3pr+r& =0\\ r(3p+1)& =0\\ 3p+1& =0\\ 3p+2& =\boxed{{\displaystyle \boxed{{\displaystyle 1}}}}\end{array}$

Jadi, 3p + 2 =1.
JAWAB: D

No. 13

Jika diketahui matriks A memenuhi persamaan $\left(\begin{array}{cc}5& 1\\ 7& 2\end{array}\right)A=\left(\begin{array}{cc}3& -2\\ -3& 1\end{array}\right)\left(\begin{array}{cc}3& 4\\ 1& 2\end{array}\right)$, maka determinan dari A^{-1} adalah

1. -2
2. -\dfrac12
3. 0

4. \dfrac12
5. 2

$\begin{array}{rl}\left(\begin{array}{cc}5& 1\\ 7& 2\end{array}\right)A& =\left(\begin{array}{cc}3& -2\\ -3& 1\end{array}\right)\left(\begin{array}{cc}3& 4\\ 1& 2\end{array}\right)\\ \left|\begin{array}{cc}5& 1\\ 7& 2\end{array}\right||A|& =\left|\begin{array}{cc}3& -2\\ -3& 1\end{array}\right|\left|\begin{array}{cc}3& 4\\ 1& 2\end{array}\right|\\ (5\cdot 2-1\cdot 7)|A|& =(3\cdot 1-(-2)\cdot (-3))(3\cdot 2-4\cdot 1)\\ (10-7)|A|& =(3-6)(6-4)\\ 3|A|& =(-3)(2)\\ 3|A|& =-6\\ |A|& =-2\end{array}$

$\begin{array}{rl}\left|A^{-1}\right|& ={\displaystyle \frac{1}{|A|}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle -{\displaystyle \frac{1}{2}}}}}}\end{array}$

Jadi, determinan dari A^{-1} adalah -\dfrac12.
JAWAB: B

No. 14

Diketahui matriks $P=\left(\begin{array}{ccc}2& -3& 6\\ 5& 0& -2\\ 1& 4& -4\end{array}\right)$. Nilai {(a_{12}-a_{31})} dari transpose P adalah

1. -4
2. -2
3. -1

4. 1
5. 11

Brainly.co.id

$P^T=\left(\begin{array}{ccc}2& 5& 1\\ -3& 0& 4\\ 6& -2& -4\end{array}\right)$

$\begin{array}{rl}{a}_{12}-a_{31}& =5-6\\ & =\boxed{{\displaystyle \boxed{{\displaystyle -1}}}}\end{array}$

Jadi, nilai {(a_{12}-a_{31})} dari transpose P adalah -1.
JAWAB: C

No. 15

Matriks X_{2\times2} yang memenuhi persamaan {AXA^{-1}=B} jika matriks $A=\left(\begin{array}{cc}2& -2\\ 1& 3\end{array}\right)$ dan $B=\left(\begin{array}{cc}3& 2\\ -1& -2\end{array}\right)$ adalah ....

1. $\left(\begin{array}{cc}3& 2\\ -1& -2\end{array}\right)$
2. $\left(\begin{array}{cc}2& -1\\ 2& 1\end{array}\right)$
3. $\left(\begin{array}{cc}3& 2\\ -1& 2\end{array}\right)$

4. $\left(\begin{array}{cc}2& -1\\ -2& -1\end{array}\right)$
5. $\left(\begin{array}{cc}2& -1\\ -2& 1\end{array}\right)$

Brainly.co.id

$\begin{array}{rl}{A}^{-1}& ={\displaystyle \frac{1}{(2)(3)-(-2)(1)}}\left(\begin{array}{cc}3& 2\\ -1& 2\end{array}\right)\\ & ={\displaystyle \frac{1}{8}}\left(\begin{array}{cc}3& 2\\ -1& 2\end{array}\right)\end{array}$

$\begin{array}{rl}AX{A}^{-1}& =B\\ AX& =BA\\ x& =A^{-1}BA\\ & ={\displaystyle \frac{1}{8}}\left(\begin{array}{cc}3& 2\\ -1& 2\end{array}\right)\left(\begin{array}{cc}3& 2\\ -1& -2\end{array}\right)\left(\begin{array}{cc}2& -2\\ 1& 3\end{array}\right)\\ & ={\displaystyle \frac{1}{8}}\left(\begin{array}{cc}7& 2\\ -5& -6\end{array}\right)\left(\begin{array}{cc}2& -2\\ 1& 3\end{array}\right)\\ & ={\displaystyle \frac{1}{8}}\left(\begin{array}{cc}16& -8\\ -16& -8\end{array}\right)\\ & =\boxed{{\displaystyle \boxed{{\displaystyle \left(\begin{array}{cc}2& -1\\ -2& -1\end{array}\right)}}}}\end{array}$

Jadi, $X=\left(\begin{array}{cc}2& -1\\ -2& -1\end{array}\right)$.
JAWAB: D

No. 16

Diketahui matriks $A=\left(\begin{array}{cc}2& -5\\ -5& 12\end{array}\right)$ dan $B=\left(\begin{array}{cc}1& -2\\ -1& 1\end{array}\right)$. Tentukan (3AB)^{-1}

Grup Telegram

$\begin{array}{rl}3AB& =3\left(\begin{array}{cc}2& -5\\ -5& 12\end{array}\right)\left(\begin{array}{cc}1& -2\\ -1& 1\end{array}\right)\\ & =3\left(\begin{array}{cc}(2)(1)+(-5)(-1)& (2)(-2)+(-5)(1)\\ (-5)(1)+(12)(-1)& (-5)(-2)+(12)(1)\end{array}\right)\\ & =3\left(\begin{array}{cc}2+5& -4-5\\ -5-12& 10+12\end{array}\right)\\ & =3\left(\begin{array}{cc}7& -9\\ -17& 22\end{array}\right)\\ & =\boxed{{\displaystyle \boxed{{\displaystyle \left(\begin{array}{cc}21& -27\\ -54& 66\end{array}\right)}}}}\end{array}$

Jadi, $(3AB{)}^{-1}=\left(\begin{array}{cc}21& -27\\ -54& 66\end{array}\right)$.

No. 17

Diketahui matriks $A=\left(\begin{array}{cc}1& 0\\ 2& 3\end{array}\right)$, $B=\left(\begin{array}{cc}-1& -3\\ 2& 0\end{array}\right)$, dan memenuhi persamaan {AX+2B=I}, dengan I adalah matriks identitas. Maka tentukan nilai determinan matriks X adalah

1. 6
2. 7
3. 8

4. 9
5. 10

SKMP Agustus 2020

$\begin{array}{rl}|A|& =(1)(3)-(0)(2)\\ & =3-0\\ & =3\end{array}$

$\begin{array}{rl}AX+2B& =I\\ AX& =I-2B\\ & =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-2\left(\begin{array}{cc}-1& -3\\ 2& 0\end{array}\right)\\ & =\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-\left(\begin{array}{cc}-2& -6\\ 4& 0\end{array}\right)\\ & =\left(\begin{array}{cc}3& 6\\ -4& 1\end{array}\right)\\ |AX|& =\left|\begin{array}{cc}3& 6\\ -4& 1\end{array}\right|\\ |A||X|& =(3)(1)-(6)(-4)\\ 3|X|& =3+24\\ & =27\\ |X|& =\boxed{{\displaystyle \boxed{{\displaystyle 9}}}}\end{array}$

Jadi, determinan matriks X adalah 9.
JAWAB: D

No. 18

Jika diketahui matriks A memenuhi persamaan $\left(\begin{array}{cc}2& 1\\ 4& 5\end{array}\right)A=\left(\begin{array}{cc}3& 1\\ 3& 2\end{array}\right)\left(\begin{array}{cc}2& 5\\ 1& 3\end{array}\right)$, maka determinan dari A^{–1} adalah

1. -2
2. -\dfrac12
3. 0

4. 1
5. 2

SKMP Juli 2020

$\begin{array}{rl}\left(\begin{array}{cc}2& 1\\ 4& 5\end{array}\right)A& =\left(\begin{array}{cc}3& 1\\ 3& 2\end{array}\right)\left(\begin{array}{cc}2& 5\\ 1& 3\end{array}\right)\\ \left|\begin{array}{cc}2& 1\\ 4& 5\end{array}\right||A|& =\left|\begin{array}{cc}3& 1\\ 3& 2\end{array}\right|\left|\begin{array}{cc}2& 5\\ 1& 3\end{array}\right|\\ 6|A|& =(3)(1)\\ 6|A|& =3\\ |A|& ={\displaystyle \frac{3}{6}}\\ & ={\displaystyle \frac{1}{2}}\end{array}$

$\begin{array}{rl}\left|A^{-1}\right|& ={\displaystyle \frac{1}{|A|}}\\ & ={\displaystyle \frac{1}{{\displaystyle \frac{1}{2}}}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 2}}}}\end{array}$

Jadi,
JAWAB: E
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No. 19

Jika matriks $A=\left(\begin{array}{cc}2& 1\\ 3& -4\end{array}\right)$, $B=\left(\begin{array}{cc}-4& -1\\ -3& 2\end{array}\right)$, dan $C=\left(\begin{array}{cc}-11& 0\\ 0& -11\end{array}\right)$, maka {(A\times B)-C} sama dengan

1. $\left(\begin{array}{cc}1& 1\\ 1& 1\end{array}\right)$
2. $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$
3. $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$

4. $\left(\begin{array}{cc}-1& -1\\ -1& -1\end{array}\right)$
5. $\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$

Grup Telegram

$\begin{array}{rl}(A\times B)-C& =\left(\begin{array}{cc}2& 1\\ 3& -4\end{array}\right)\left(\begin{array}{cc}-4& -1\\ -3& 2\end{array}\right)-\left(\begin{array}{cc}-11& 0\\ 0& -11\end{array}\right)\\ & =\left(\begin{array}{cc}-11& 0\\ 0& -11\end{array}\right)-\left(\begin{array}{cc}-11& 0\\ 0& -11\end{array}\right)\\ & =\boxed{{\displaystyle \boxed{{\displaystyle \left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)}}}}\end{array}$

Jadi, $(A\times B)-C=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)$.
JAWAB: E

No. 20

Jika diketahui $\left(\begin{array}{c}x+3\\ 4-y\end{array}\right)=\left(\begin{array}{c}7\\ 5\end{array}\right)$, Nilai dari x+y= ....

Grup Telegram

$\begin{array}{rl}x+3& =7\\ x& =4\end{array}$

$\begin{array}{rl}4-y& =5\\ -y& =1\\ y& =-1\end{array}$

$\begin{array}{rl}x+y& =4+(-1)\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 3}}}}\end{array}$

Jadi, x+y=3.

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