MATRIKS

 

Soal Nomor 1 (Soal UN SMA Semuan Jurusan)

Diketahui matriks $A=\left(\begin{array}{cc}3& y\\ 5& -1\end{array}\right),B=\left(\begin{array}{cc}x& 5\\ -3& 6\end{array}\right)$, dan $C=\left(\begin{array}{cc}-3& -1\\ y& 9\end{array}\right)$. Jika $A+B-C=\left(\begin{array}{cc}8& 5x\\ -x& -4\end{array}\right)$, maka nilai $x+2xy+y$ adalah $\cdots \cdot$
A. $8$                      C. $18$                     E. $22$
B. $12$                    D. $20$     

Pembahasan

Diketahui:
$\begin{array}{cc}A& =\left(\begin{array}{cc}3& y\\ 5& -1\end{array}\right)\\ B& =\left(\begin{array}{cc}x& 5\\ -3& 6\end{array}\right)\\ C& =\left(\begin{array}{cc}-3& -1\\ y& 9\end{array}\right)\end{array}$
Dengan demikian, 
$$\begin{array}{cc}A+B-C& =\left(\begin{array}{cc}8& 5x\\ -x& -4\end{array}\right)\\ \left(\begin{array}{cc}3& y\\ 5& -1\end{array}\right)+\left(\begin{array}{cc}x& 5\\ -3& 6\end{array}\right)-\left(\begin{array}{cc}-3& -1\\ y& 9\end{array}\right)& =\left(\begin{array}{cc}8& 5x\\ -x& -4\end{array}\right)\\ \left(\begin{array}{cc}3+x-(-3)& y+5-(-1)\\ 5+(-3)-y& -1+6-9\end{array}\right)& =\left(\begin{array}{cc}8& 5x\\ -x& -4\end{array}\right)\\ \left(\begin{array}{cc}x+6& y+6\\ 2-y& -4\end{array}\right)& =\left(\begin{array}{cc}8& 5x\\ -x& -4\end{array}\right)\end{array}$$Persamaan terakhir menunjukkan bahwa
$x+6=8$, sehingga $x=2$, dan
$y+6=5x$, berarti $y+6=5\left(2\right)=10$, sehingga didapat $y=4$. 
Jadi, nilai dari $x+2xy+y$ adalah
$2+2\left(2\right)\left(4\right)+4=22$
(Jawaban E)

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Soal Nomor 2 (Soal UN SMA Semua Jurusan)
Diketabui matriks $A=\left(\begin{array}{cc}a+2& 1-3b\\ -1& -6\end{array}\right),B=\left(\begin{array}{cc}2a& b-3\\ -1& 2\end{array}\right)$, dan $C=\left(\begin{array}{cc}5& 6\\ -2& -4\end{array}\right)$. Jika $A+B=C$, maka nilai $a+b$ adalah $\cdots \cdot$
A. $-6$                        C. $-2$                   E. $2$
B. $-3$                        D. $1$       

Pembahasan

Diketahui:
$\begin{array}{cc}A& =\left(\begin{array}{cc}a+2& 1-3b\\ -1& -6\end{array}\right)\\ B& =\left(\begin{array}{cc}2a& b-3\\ -1& 2\end{array}\right)\\ C& =\left(\begin{array}{cc}5& 6\\ -2& -4\end{array}\right)\end{array}$
Dengan demikian, 
$$\begin{array}{cc}A+B& =C\\ \left(\begin{array}{cc}a+2& 1-3b\\ -1& -6\end{array}\right)+\left(\begin{array}{cc}2a& b-3\\ -1& 2\end{array}\right)& =\left(\begin{array}{cc}5& 6\\ -2& -4\end{array}\right)\\ \left(\begin{array}{cc}3a+2& -2b-2\\ -2& -4\end{array}\right)& =\left(\begin{array}{cc}5& 6\\ -2& -4\end{array}\right)\end{array}$$Persamaan terakhir menunjukkan bahwa
$3a+2=5⟺3a=3⟺a=1$
dan
$-2b-2=6⟺-2b=8⟺b=-4$
Jadi, nilai dari $a+b$ adalah $1+(-4)=-3$
(Jawaban B)

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Soal Nomor 3 (Soal UN SMA Semua Jurusan)

Diketahui matriks $A=\left(\begin{array}{cc}3& -1\\ 2m& -3\end{array}\right),B=\left(\begin{array}{cc}n+1& 3\\ m-n& 0\end{array}\right)$, dan $C=\left(\begin{array}{cc}5& -4\\ 2& -3\end{array}\right)$. Jika $C^T$ adalah transpos dari $C$ dan $A+B=C^T$, nilai dari $3m+2n=\cdots \cdot$
A. $-25$                 C. $-11$               E. $-1$
B. $-14$                 D. $-7$     

Pembahasan

Diketahui:
$\begin{array}{cc}A& =\left(\begin{array}{cc}3& -1\\ 2m& -3\end{array}\right)\\ B& =\left(\begin{array}{cc}n+1& 3\\ m-n& 0\end{array}\right)\\ C& =\left(\begin{array}{cc}5& -4\\ 2& -3\end{array}\right)\\ C^T& =\left(\begin{array}{cc}5& 2\\ -4& -3\end{array}\right)\end{array}$
Dengan demikian, 
$$\begin{array}{cc}A+B& =C^T\\ \left(\begin{array}{cc}3& -1\\ 2m& -3\end{array}\right)+\left(\begin{array}{cc}n+1& 3\\ m-n& 0\end{array}\right)& =\left(\begin{array}{cc}5& 2\\ -4& -3\end{array}\right)\\ \left(\begin{array}{cc}n+4& 2\\ 3m-n& -3\end{array}\right)& =\left(\begin{array}{cc}5& 2\\ -4& -3\end{array}\right)\end{array}$$Persamaan terakhir menunjukkan bahwa
$n+4=5⟺n=1$
dan
$\begin{array}{cc}3m-n& =-4\\ ⟹3m-1& =-4⟺m=-1\end{array}$
Jadi, nilai dari $3m+2n$ adalah $3(-1)+2\left(1\right)=-1$
(Jawaban E)

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Soal Nomor 4 (Soal UN SMA Semuan Jurusan)
Diketahui persamaan matriks $3\left(\begin{array}{cc}5& x\\ y& 4\end{array}\right)+\left(\begin{array}{cc}-6& x-4\\ 3-y& -7\end{array}\right)=\left(\begin{array}{cc}9& 8\\ 13& 5\end{array}\right)$
Nilai $2x-y$ adalah $\cdots \cdot$
A. $-2$                       C. $1$                     E. $5$
B. $-1$                       D. $3$       

Pembahasan

Dari persamaan matriks yang diberikan itu, dapat ditulis
$$\begin{array}{cc}\left(\begin{array}{cc}15& 3x\\ 3y& 12\end{array}\right)+\left(\begin{array}{cc}-6& x-4\\ 3-y& -7\end{array}\right)& =\left(\begin{array}{cc}9& 8\\ 13& 5\end{array}\right)\\ \left(\begin{array}{cc}9& 4x-4\\ 2y+3& 5\end{array}\right)& =\left(\begin{array}{cc}9& 8\\ 13& 5\end{array}\right)\end{array}$$Persamaan terakhir menunjukkan bahwa
$4x-4=8⟺4x=12⟺x=3$
dan
$2y+3=13⟺2y=10⟺y=5$
Jadi, nilai dari $2x-y$ adalah $2\left(3\right)-5=1$
(Jawaban C)

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Soal Nomor 5 (Soal UN SMA Semuan Jurusan)
Diketahui persamaan matriks$$2\left(\begin{array}{cc}a& 2\\ -3& 1\end{array}\right)+\left(\begin{array}{cc}4& -1\\ 0& b\end{array}\right)=\left(\begin{array}{cc}3& 2\\ c& 4\end{array}\right)\left(\begin{array}{cc}2& d\\ 1& 3\end{array}\right)$$Nilai dari $a+b+c+d=\cdots \cdot$
A. $11$                    C. $15$                    E. $19$
B. $13$                    D. $17$       

Pembahasan

Dari persamaan matriks di atas, diperoleh
$$\begin{array}{cc}\left(\begin{array}{cc}2a& 4\\ -6& 2\end{array}\right)+\left(\begin{array}{cc}4& -1\\ 0& b\end{array}\right)& =\left(\begin{array}{cc}3\left(2\right)+2\left(1\right)& 3d+2\left(3\right)\\ 2c+4\left(1\right)& cd+4\left(3\right)\end{array}\right)\\ \left(\begin{array}{cc}2a+4& 3\\ -6& 2+b\end{array}\right)& =\left(\begin{array}{cc}8& 3d+6\\ 2c+4& cd+12\end{array}\right)\end{array}$$Dari persamaan terakhir, didapat
$2a+4=8⟺2a=4⟺a=2$
$3=3d+6⟺3d=-3⟺d=-1$
$-6=2c+4⟺2c=-10⟺c=-5$
dan
$\begin{array}{cc}& 2+b=cd+12\\ & ⟹2+b=(-5)(-1)+12=17\\ & ⟺b=15\end{array}$
Jadi, nilai dari $a+b+c+d$ adalah $2+15+(-5)+(-1)=11$
(Jawaban A)