Exercise Zone : Matriks

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Tipe:

Standar SBMPTN HOTS

No.

Jika matriks {A\cdot B=$$\left(\begin{array}{cc}-8& -5\\ 4& 8\end{array}\right)$$} dan {\det A=4}, maka {\det\left(4BA^{-1}\right)} adalah....

1. -46
2. -45
3. -44

4. -43
5. -42

$$\begin{array}{rl}A\cdot B& =\left(\begin{array}{cc}-8& -5\\ 4& 8\end{array}\right)\\ |AB|& =(-8)(8)-(-5)(4)\\ |A||B|& =-64+20\\ 4|B|& =-44\\ |B|& =-11\end{array}$$

$$\begin{array}{rl}\left|4B{A}^{-1}\right|& =4^2|B|\left|A^{-1}\right|\\ & =16(-11)\left({\displaystyle \frac{1}{|A|}}\right)\\ & =-176\left({\displaystyle \frac{1}{4}}\right)\\ & =-44\end{array}$$

Jadi, \det\left(4BA^{-1}\right)=-44. JAWAB: C

No.

Jika {A=$$\left(\begin{array}{ccc}-1& -1& 0\\ -1& 1& 2\end{array}\right)$$}, {B=$$\left(\begin{array}{cc}-7& x\\ -6& y\\ -2& z\end{array}\right)$$} dan {AB=$$\left(\begin{array}{cc}13& 2\\ -3& 12\end{array}\right)$$}, maka nilai {z-x} adalah ....

$$\begin{array}{rl}AB& =\left(\begin{array}{cc}13& 2\\ -3& 12\end{array}\right)\\ \left(\begin{array}{ccc}-1& -1& 0\\ -1& 1& 2\end{array}\right)\left(\begin{array}{cc}-7& x\\ -6& y\\ -2& z\end{array}\right)& =\left(\begin{array}{cc}13& 2\\ -3& 12\end{array}\right)\\ \left(\begin{array}{cc}13& -x-y\\ -3& -x+y+2z\end{array}\right)& =\left(\begin{array}{cc}13& 2\\ -3& 12\end{array}\right)\end{array}$$

$$\begin{array}{rl}-x-y& =2\\ -x+y+2z& =12& +\\ -2x+2z& =14\\ 2z-2x& =14\\ z-x& =\boxed{{\displaystyle \boxed{{\displaystyle 7}}}}\end{array}$$

Jadi, z-x=7.

No.

Diketahui matriks {A=$$\left(\begin{array}{cc}-4& -4\\ 7& -7\end{array}\right)$$} dan {B=$$\left(\begin{array}{cc}1& y\\ x& 3\end{array}\right)$$}. Jika determinan AB adalah -336 maka xy= ....

SUMBER

$$\begin{array}{rl}|AB|& =-336\\ |A||B|& =-336\\ (56)(3-xy)& =-336\\ 3-xy& =-6\\ xy& =\boxed{{\displaystyle \boxed{{\displaystyle 9}}}}\end{array}$$

Jadi, xy=9.

No.

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

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

Jadi, p=-\dfrac12.

No.

Matriks A yang memenuhi {$$\left(\begin{array}{cc}2& k\\ 1& 0\end{array}\right)$$A=$$\left(\begin{array}{cc}2& 4k\\ 1& 0\end{array}\right)$$} 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=$$\left(\begin{array}{cc}1& 0\\ 0& 4\end{array}\right)$$.

No.

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

1. $$\left(\begin{array}{cc}-2& 8\\ -8& 3\end{array}\right)$$
2. $$\left(\begin{array}{cc}1& 7\\ -6& 3\end{array}\right)$$
3. $$\left(\begin{array}{cc}2& 7\\ 9& -3\end{array}\right)$$

4. $$\left(\begin{array}{cc}2& 7\\ 9& 3\end{array}\right)$$
5. $$\left(\begin{array}{cc}2& 9\\ 7& -3\end{array}\right)$$

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

Jadi, 2K-L+M=$$\left(\begin{array}{cc}2& 7\\ 9& -3\end{array}\right)$$.

No.

Diketahui matriks {A=$$\left(\begin{array}{cc}-1& 2\\ 3& -2\end{array}\right)$$} dan {B=$$\left(\begin{array}{cc}1& -2\\ 3& 0\end{array}\right)$$}. Jika matriks {C=AB}, maka determinan matriks C= ....

1. 18
2. 12
3. -24

4. -30
5. -36

$$\begin{array}{rl}C& =AB\\ |C|& =|A||B|\\ & =[(-1)(-2)-(2)(3)]\cdot [(1)(0)-(-2)(3)]\\ & =[2-6][0-(-6)]\\ & =(-4)(6)\\ & =-24\end{array}$$

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

No.

Diketahui persamaan matriks:
{2$$\left(\begin{array}{cc}x& 2\\ 6& 6\end{array}\right)$$+$$\left(\begin{array}{cc}2& 4\\ -2& 0\end{array}\right)$$=$$\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)$$$$\left(\begin{array}{cc}-1& 2\\ 3& y\end{array}\right)$$}
Nilai x-y=

1. -2
2. 2
3. 0

4. 1
5. -1

$$\begin{array}{rl}2\left(\begin{array}{cc}x& 2\\ 6& 6\end{array}\right)+\left(\begin{array}{cc}2& 4\\ -2& 0\end{array}\right)& =\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)\left(\begin{array}{cc}-1& 2\\ 3& y\end{array}\right)\\ \left(\begin{array}{cc}2x& 4\\ 12& 12\end{array}\right)+\left(\begin{array}{cc}2& 4\\ -2& 0\end{array}\right)& =\left(\begin{array}{cc}8& 3y+2\\ 10& 4y+4\end{array}\right)\\ \left(\begin{array}{cc}2x+2& 8\\ 10& 12\end{array}\right)& =\left(\begin{array}{cc}8& 3y+2\\ 10& 4y+4\end{array}\right)\end{array}$$

Jadi, x-y=1. JAWAB: D

No.

Misalkan A^T adalah transpose matriks A dan {I=$$\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$$}. Jika {A=$$\left(\begin{array}{cc}8& 0\\ a& b\end{array}\right)$$} sehingga {5A=3A^T+16I}, maka nilai {10a+4b} adalah

1. 32
2. 34
3. 35

4. 36
5. 40

{A^T=$$\left(\begin{array}{cc}8& a\\ 0& b\end{array}\right)$$}

$$\begin{array}{rl}5A& =3A^T+16I\\ 5\left(\begin{array}{cc}8& 0\\ a& b\end{array}\right)& =3\left(\begin{array}{cc}8& a\\ 0& b\end{array}\right)+16\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\\ \left(\begin{array}{cc}40& 0\\ 5a& 5b\end{array}\right)& =\left(\begin{array}{cc}24& 3a\\ 0& 3b\end{array}\right)+\left(\begin{array}{cc}16& 0\\ 0& 16\end{array}\right)\\ \left(\begin{array}{cc}40& 0\\ 5a& 5b\end{array}\right)& =\left(\begin{array}{cc}40& 3a\\ 0& 3b+16\end{array}\right)\end{array}$$

$$\begin{array}{rl}0& =3a\\ a& =0\end{array}$$

$$\begin{array}{rl}5b& =3b+16\\ 2b& =16\\ 4b& =32\end{array}$$

$$\begin{array}{rl}10a+4b& =10(0)+32\\ & =\boxed{{\displaystyle \boxed{{\displaystyle 32}}}}\end{array}$$

Jadi, 10a+4b=32. JAWAB: A

No.

Diketahui matriks A berordo 2\times2 dan {B=$$\left(\begin{array}{cc}-3& 5\\ -1& 2\end{array}\right)$$} dan {C=$$\left(\begin{array}{cc}4& 5\\ 2& 3\end{array}\right)$$}. Jika A memenuhi {B\cdot A=C}, maka \det\left(2A^{-1}\right) adalah

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

4. \dfrac12
5. 2

$$\begin{array}{rl}detB& =(-3)(2)-(5)(-1)\\ & =-6-(-5)\\ & =-1\end{array}$$

$$\begin{array}{rl}detC& =(4)(3)-(2)(5)\\ & =12-10\\ & =2\end{array}$$

$$\begin{array}{rl}B\cdot A& =C\\ A& =B^{-1}\cdot C\\ detA& =det(B^{-1}\cdot C)\\ & ={\displaystyle \frac{1}{detB}}\cdot detC\\ & ={\displaystyle \frac{1}{-1}}\cdot 2\\ & =-2\end{array}$$

$$\begin{array}{rl}det\left(2A^{-1}\right)& =2^{2}det{A}^{-1}\\ & =4\cdot {\displaystyle \frac{1}{detA}}\\ & =4\cdot {\displaystyle \frac{1}{-2}}\\ & =\boxed{{\displaystyle \boxed{{\displaystyle -2}}}}\end{array}$$

Jadi, \det\left(2A^{-1}\right)=-2. JAWAB: A