Exercise Zone : Matriks

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

• 1
• 2

---

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

---

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.

---

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.

---

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.

---

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.

---

No. 6

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

\(\eqalign{ \pmatrix{2&k\\1&0}A&=\pmatrix{2&4k\\1&0}\\ A&=\pmatrix{2&k\\1&0}^{-1}\pmatrix{2&4k\\1&0}\\ &=\dfrac1{2\cdot0-k\cdot1}\pmatrix{0&-k\\-1&2}\pmatrix{2&4k\\1&0}\\ &=\dfrac1{-k}\pmatrix{-k&0\\0&-4k}\\ &=\pmatrix{1&0\\0&4} }\)

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

---

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 ....

\(\eqalign{ A^{-1}B^{-1}C^{-1}D^{-1}E^{-1}&=M\\ EDCBA&=M^{-1}\\ DCBA&=E^{-1}M^{-1}\\ CBA&=D^{-1}E^{-1}M^{-1}\\ CB&=D^{-1}E^{-1}M^{-1}A^{-1}\\ C&=D^{-1}E^{-1}M^{-1}A^{-1}B^{-1} }\)

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

---

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. \(\begin{pmatrix}2&9\\7&-3\end{pmatrix}\)

\(\eqalign{ 2K-L+M&=2\pmatrix{-1&4\\3&-2}-\pmatrix{5&1\\3&-2}+\pmatrix{9&0\\6&-1}\\ &=\pmatrix{-2&8\\6&-4}-\pmatrix{5&1\\3&-2}+\pmatrix{9&0\\6&-1}\\ &=\boxed{\boxed{\color{blue}\pmatrix{2&7\\9&-3}}} }\)

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

---

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

---

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

\(\eqalign{ A+B&=C\\ \pmatrix{1&x&-1\\3&3&x}+\pmatrix{2&y&3\\4&5&x}&=\pmatrix{3&x&2\\6&8&2}\\ \pmatrix{3&x+y&2\\7&8&2x}&=\pmatrix{3&x&2\\6&8&2} }\)

\(\eqalign{ x+y&=x\\ y&=0 }\)

\(\eqalign{ 2x&=2\\ x&=1 }\)

\(\eqalign{ x+y&=1+0\\ &=1 }\)

Jadi, x+y=1.
JAWAB: D

• 1
• 2

Share this post