Exercise Zone : Matriks

Berikut ini adalah kumpulan soal mengenai Matriks tingkat dasar. Jika ada jawaban yang salah, mohon dikoreksi melalui komentar. Terima kasih.

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No. 1

Jika matriks ${A\cdot B=\begin{pmatrix}5&6\\2&3\end{pmatrix}}$ dan ${\det A=3}$, maka ${\det\left(3BA^{-1}\right)}$ adalah....

1. $1$
2. $2$
3. $3$

4. $4$
5. $5$

$\begin{aligned}A\cdot B&=\begin{pmatrix}5&6\\2&3\end{pmatrix}\\[8pt]|AB|&=(5)(3)-(2)(6)\\|A||B|&=15-12\\3|B|&=3\\|B|&=1\end{aligned}$

$\begin{aligned}\left|3BA^{-1}\right|&=3^2|B|\left|A^{-1}\right|\\&=9(1)\left(\dfrac1{|A|}\right)\\&=9\left(\dfrac13\right)\\&=3\end{aligned}$

Jadi, $\det\left(3BA^{-1}\right)=3$.
JAWAB: C

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No. 2

Jika ${A=\begin{pmatrix}-1&-1&0\\-1&1&2\end{pmatrix}}$, ${B=\begin{pmatrix}-1&x\\1&y\\0&z\end{pmatrix}}$ dan ${AB=\begin{pmatrix}0&2\\2&4\end{pmatrix}}$, maka nilai ${z-x}$ adalah ....

$\begin{aligned}AB&=\begin{pmatrix}0&2\\2&4\end{pmatrix}\\\begin{pmatrix}-1&-1&0\\-1&1&2\end{pmatrix}\begin{pmatrix}-1&x\\1&y\\0&z\end{pmatrix}&=\begin{pmatrix}0&2\\2&4\end{pmatrix}\\\begin{pmatrix}0&-x-y\\2&-x+y+2z\end{pmatrix}&=\begin{pmatrix}0&2\\2&4\end{pmatrix}\end{aligned}$

$\begin{aligned}-x-y&=2\\-x+y+2z&=4&\qquad+\\\hline-2x+2z&=6\\2z-2x&=6\\z-x&=3\end{aligned}$

Jadi, $z-x=3$.

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No. 3

Diketahui matriks ${A=\begin{pmatrix}1&2\\3&4\end{pmatrix}}$ dan ${B=\begin{pmatrix}1&y\\x&3\end{pmatrix}}$. Jika determinan $AB$ adalah $10$ maka $xy=$ ....

$\begin{aligned}|AB|&=10\\|A||B|&=10\\(-2)(3-xy)&=10\\3-xy&=-5\\xy&=8\end{aligned}$

Jadi, $xy=8$.

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No. 4

Diberikan matriks ${P=\begin{pmatrix}2&-1\\4&3\end{pmatrix}}$ dan ${Q=\begin{pmatrix}2r&1\\r&p+1\end{pmatrix}}$ dengan ${r\neq0}$ dan ${p\neq0}$. Matriks $PQ$ tidak mempunyai invers apabila nilai $p=$ ....

Matriks $PQ$ tidak mempunyai invers maka:

$\begin{aligned}|PQ|&=0\\|P||Q|&=0\\((2)(3)-(-1)(4))((2r)(p+1)-(1)(r))&=0\\(10)(2pr+2r-r)&=0\\2pr+r&=0\\r(2p+1)&=0\\2p+1&=0\\2p&=-1\\p&=-\dfrac12\end{aligned}$

Jadi, $p=-\dfrac12$.

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No. 5

Jika ${P=\begin{pmatrix}1&2\\1&3\end{pmatrix}}$ dan ${\begin{pmatrix}x&y\\-z&z\end{pmatrix}=2P^{-1}}$ dengan $P^{-1}$ menyatakan invers matriks $P$, maka ${x+y=}$ ....

$\begin{aligned}|P|&=(1)(3)-(2)(1)\\&=1\end{aligned}$

$\begin{aligned}P^{-1}&=\dfrac1{|P|}\begin{pmatrix}3&-2\\-1&1\end{pmatrix}\\&=\dfrac11\begin{pmatrix}3&-2\\-1&1\end{pmatrix}\\&=\begin{pmatrix}3&-2\\-1&1\end{pmatrix}\end{aligned}$

$\begin{aligned}\begin{pmatrix}x&y\\-z&z\end{pmatrix}&=2P^{-1}\\\begin{pmatrix}x&y\\-z&z\end{pmatrix}&=2\begin{pmatrix}3&-2\\-1&1\end{pmatrix}\\\begin{pmatrix}x&y\\-z&z\end{pmatrix}&=\begin{pmatrix}6&-4\\-2&2\end{pmatrix}\end{aligned}$
$x=6$ dan $y=-4$

$\begin{aligned}x+y&=6+(-4)\\&=2\end{aligned}$

Jadi, $x+y=2$.

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No. 6

Matriks $A$ yang memenuhi ${\begin{pmatrix}2&k\\1&0\end{pmatrix}A=\begin{pmatrix}2&4k\\1&0\end{pmatrix}}$ adalah ....

$\begin{array}{rl}\left(\begin{array}{cc}2& k\\ 1& 0\end{array}\right)A& =\left(\begin{array}{cc}2& 4k\\ 1& 0\end{array}\right)\\ A& ={\left(\begin{array}{cc}2& k\\ 1& 0\end{array}\right)}^{-1}\left(\begin{array}{cc}2& 4k\\ 1& 0\end{array}\right)\\ & ={\displaystyle \frac{1}{2\cdot 0-k\cdot 1}}\left(\begin{array}{cc}0& -k\\ -1& 2\end{array}\right)\left(\begin{array}{cc}2& 4k\\ 1& 0\end{array}\right)\\ & ={\displaystyle \frac{1}{-k}}\left(\begin{array}{cc}-k& 0\\ 0& -4k\end{array}\right)\\ & =\left(\begin{array}{cc}1& 0\\ 0& 4\end{array}\right)\end{array}$

Jadi, $A=\begin{pmatrix}1&0\\0&4\end{pmatrix}$.

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No. 7

Jika $A$, $B$, $C$, $D$, $E$, dan $M$ adalah matriks persegi yang berukuran sama dan memenuhi ${A^{-1}B^{-1}C^{-1}D^{-1}E^{-1}=M}$, maka matriks $C$ adalah ....

$\begin{array}{rl}{A}^{-1}{B}^{-1}{C}^{-1}{D}^{-1}{E}^{-1}& =M\\ EDCBA& =M^{-1}\\ DCBA& =E^{-1}{M}^{-1}\\ CBA& =D^{-1}{E}^{-1}{M}^{-1}\\ CB& =D^{-1}{E}^{-1}{M}^{-1}{A}^{-1}\\ C& =D^{-1}{E}^{-1}{M}^{-1}{A}^{-1}{B}^{-1}\end{array}$

Jadi, $C=D^{-1}E^{-1}M^{-1}A^{-1}B^{-1}$.

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No. 8

Diketahui matriks $K=\begin{pmatrix}-1&4\\3&-2\end{pmatrix}$, $L=\begin{pmatrix}5&1\\3&-2\end{pmatrix}$ dan $M=\begin{pmatrix}9&0\\6&-1\end{pmatrix}$. Matriks ${2K-L+M}$ adalah....

1. $\begin{pmatrix}-2&8\\-8&3\end{pmatrix}$
2. $\begin{pmatrix}1&7\\-6&3\end{pmatrix}$
3. $\begin{pmatrix}2&7\\9&-3\end{pmatrix}$

4. $\begin{pmatrix}2&7\\9&3\end{pmatrix}$
5. $\begin{pmatrix}2&9\\7&-3\end{pmatrix}$

$\begin{array}{rl}2K-L+M& =2\left(\begin{array}{cc}-1& 4\\ 3& -2\end{array}\right)-\left(\begin{array}{cc}5& 1\\ 3& -2\end{array}\right)+\left(\begin{array}{cc}9& 0\\ 6& -1\end{array}\right)\\ & =\left(\begin{array}{cc}-2& 8\\ 6& -4\end{array}\right)-\left(\begin{array}{cc}5& 1\\ 3& -2\end{array}\right)+\left(\begin{array}{cc}9& 0\\ 6& -1\end{array}\right)\\ & =\boxed{{\displaystyle \boxed{{\displaystyle \left(\begin{array}{cc}2& 7\\ 9& -3\end{array}\right)}}}}\end{array}$

Jadi, $2K-L+M=\begin{pmatrix}2&7\\9&-3\end{pmatrix}$.
JAWAB: C

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No. 9

Diketahui matriks $A=\begin{pmatrix}-1&2\\3&-2\end{pmatrix}$ dan $B=\begin{pmatrix}1&-2\\3&0\end{pmatrix}$. Jika matriks ${C=AB}$, maka determinan matriks $C=$

1. $18$
2. $12$
3. $-24$

4. $-30$
5. $-36$

\(\eqalign{ C&=AB\ |C|&=|A||B|\ &=\left[(-1)(-2)-(2)(3)\right]\cdot[(1)(0)-(-2)(3)]\ &=[2-6][0-(-6)]\ &=(-4)(6)\ &=-24 }$

Jadi, determinan matriks $C$ adalah $-24$.
JAWAB: C

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No. 10

Diketahui matriks $A=\begin{pmatrix}1&x&-1\\3&3&x\end{pmatrix}$, $B=\begin{pmatrix}2&y&3\\4&5&x\end{pmatrix}$, dan $C=\begin{pmatrix}3&x&2\\6&8&2\end{pmatrix}$. Jika ${A+B=C}$, maka ${x+y=}$ ....

1. $-2$
2. $-1$
3. $0$

4. $1$
5. $4$

$\begin{array}{rl}A+B& =C\\ \left(\begin{array}{ccc}1& x& -1\\ 3& 3& x\end{array}\right)+\left(\begin{array}{ccc}2& y& 3\\ 4& 5& x\end{array}\right)& =\left(\begin{array}{ccc}3& x& 2\\ 6& 8& 2\end{array}\right)\\ \left(\begin{array}{ccc}3& x+y& 2\\ 7& 8& 2x\end{array}\right)& =\left(\begin{array}{ccc}3& x& 2\\ 6& 8& 2\end{array}\right)\end{array}$

$\begin{array}{rl}x+y& =x\\ y& =0\end{array}$

$\begin{array}{rl}2x& =2\\ x& =1\end{array}$

$\begin{array}{rl}x+y& =1+0\\ & =1\end{array}$

Jadi, $x+y=1$.
JAWAB: D

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