# Determinan Matriks Persegi Berordo 3 x 3 Menggunakan Aturan Sarrus
Misal $A=\begin{pmatrix} a_{11}\; &a_{12}\; &a_{13} \\ a_{21}\; &a_{22} \; &a_{23} \\ a_{31}\;&a_{32}\; & a_{33} \end{pmatrix}$
$\begin{array}{lcl} detA=|A|=\begin{vmatrix} a_{11}\; &a_{12}\; &a_{13} \\ a_{21}\; &a_{22} \; &a_{23} \\ a_{31}\;&a_{32}\; & a_{33} \end{vmatrix}\begin{matrix} a_{11}\; &a_{12} \\ a_{21}\; & a_{22}\\ a_{31}\; & a_{32} \end{matrix}\\ \; \; \; \; \; \; \; \; =a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12} \end{array}$
Contoh:
Hitunglah determinan matriks $\begin{array}{lcl} A=\begin{pmatrix} 1\; & 0\; &1 \\ -1\; & 3\; &0 \\ 1\; &0\; &2 \end{pmatrix} \end{array}$
Pembahasan:
$\begin{array}{lcl} detA=\begin{vmatrix} 1\; & 0\; &1 \\ -1\; & 3\; &0 \\ 1\; &0\; &2 \end{vmatrix}\begin{matrix} 1\; &0 \\ -1\; & 3\\ 1\; &0 \end{matrix}\\ \; \; \; \; \; \; \; \; =\left ( 1 \right )\left ( 3 \right )\left ( 2 \right )+\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )+\left ( 1 \right )\left ( -1 \right )\left ( 0 \right )-\left ( 1 \right )\left ( 3 \right )\left ( 1 \right )-\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )-\left ( 2 \right )\left ( -1 \right )\left ( 0 \right )\\ \; \; \; \; \; \; \; \; =6+0+0-3-0-0\\ \; \; \; \; \; \; \; \; =3 \end{array}$
# Determinan Matriks Berordo 3 x 3 Menggunakan Kofaktor
Menentukan minor $(M)$ dan kofaktor $(C)$ dari matriks berordo 3
Misal matriks $\begin{array}{lcl} A=\begin{pmatrix} a_{11}\; &a_{12}\; &a_{13} \\ a_{21}\; &a_{22}\; & a_{23}\\ a_{31}\; &a_{32}\; & a_{33} \end{pmatrix} \end{array}$
Minor dan kofaktor dari matriks $A$
1. $\begin{array}{lcl} M_{11}=\begin{vmatrix} a_{22 }\; & a_{23}\\ a_{32}\; & a_{33} \end{vmatrix}\\ C_{11}=\left ( -1 \right )^{1+1}M_{11}=\begin{vmatrix} a_{22 }\; & a_{23}\\ a_{32}\; & a_{33} \end{vmatrix}\\ \end{array}$
2. $\begin{array}{lcl} M_{12}=\begin{vmatrix} a_{21 }\; & a_{23}\\ a_{31}\; & a_{33} \end{vmatrix}\\ C_{12}=\left ( -1 \right )^{1+2}M_{12}=-\begin{vmatrix} a_{21 }\; & a_{23}\\ a_{31}\; & a_{33} \end{vmatrix}\\ \end{array}$
3. $\begin{array}{lcl} M_{13}=\begin{vmatrix} a_{21 }\; & a_{22}\\ a_{31}\; & a_{32} \end{vmatrix}\\ C_{13}=\left ( -1 \right )^{1+3}M_{13}=\begin{vmatrix} a_{21 }\; & a_{22}\\ a_{31}\; & a_{32} \end{vmatrix}\\ \end{array}$
4. $\begin{array}{lcl} M_{21}=\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{32}\; & a_{33} \end{vmatrix}\\ C_{21}=\left ( -1 \right )^{2+1}M_{21}=-\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{32}\; & a_{33} \end{vmatrix}\\ \end{array}$
5. $\begin{array}{lcl} M_{22}=\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{31}\; & a_{33} \end{vmatrix}\\ C_{22}=\left ( -1 \right )^{2+2}M_{21}=\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{31}\; & a_{33} \end{vmatrix}\\ \end{array}$
6. $\begin{array}{lcl} M_{23}=\begin{vmatrix} a_{11 }\; & a_{12}\\ a_{31}\; & a_{32} \end{vmatrix}\\ C_{23}=\left ( -1 \right )^{2+3}M_{23}=-\begin{vmatrix} a_{11 }\; & a_{12}\\ a_{31}\; & a_{32} \end{vmatrix}\\ \end{array}$
7. $\begin{array}{lcl} M_{31}=\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{22}\; & a_{23} \end{vmatrix}\\ C_{31}=\left ( -1 \right )^{3+1}M_{31}=\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{22}\; & a_{23} \end{vmatrix}\\ \end{array}$
8. $\begin{array}{lcl} M_{32}=\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{21}\; & a_{23} \end{vmatrix}\\ C_{32}=\left ( -1 \right )^{3+2}M_{32}=-\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{21}\; & a_{23} \end{vmatrix}\\ \end{array}$
Determinan dari matriks $A$ (pilih salah satu saja)
$\begin{array}{lcl} 1.\; detA=|A|=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\\ 2.\; detA=|A|=a_{21}C_{21}+a_{22}C_{22}+a_{23}C_{23}\\ 3.\; detA=|A|=a_{31}C_{31}+a_{32}C_{32}+a_{33}C_{33}\\ 4.\; detA=|A|=a_{11}C_{11}+a_{21}C_{21}+a_{31}C_{31}\\ 5.\; detA=|A|=a_{12}C_{12}+a_{22}C_{22}+a_{32}C_{32}\\ 6.\; detA=|A|=a_{13}C_{13}+a_{23}C_{23}+a_{33}C_{33}\\ \ \end{array}$
Contoh:
Hitunglah determinan matriks menggunakan kofaktor $\begin{array}{lcl} A=\begin{pmatrix} 1 \; & 0\; &1 \\ -1\; &3\; &0 \\ 1\; & 0\; &2 \end{pmatrix} \ \end{array}$
Pembahasan:
Untuk menghitung determinan matriks $A$ dengan menggunakan kofaktor,
kita menggunakan rumus 1 (bisa dipilih)
$\begin{array}{lcl} a_{11}=1\\ C_{11}=\begin{vmatrix} 3\; &0 \\ 0\; & 2 \end{vmatrix}\\ C_{11}=6\\\\ a_{12}=0\\ C_{12}=-\begin{vmatrix} -1\; &0 \\ 1\; & 2 \end{vmatrix}\\ C_{12}=2\\ \ \end{array}$
$\begin{array}{lcl} a_{13}=1\\ C_{13}=\begin{vmatrix} -1\; &3 \\ 1\; & 0 \end{vmatrix}\\ C_{13}=0-3\\ C_{13}=-3\\\\ detA=|A|=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\\ \; \; \; \; \; \; \; \; =\left ( 1 \right )\left ( 6 \right )+\left ( 0 \right )\left ( 2 \right )+\left ( 1 \right )\left ( -3 \right )\\ \; \; \; \; \; \; \; \; =6+0-3\\ \; \; \; \; \; \; \; \; =3 \ \end{array}$
# Invers Matriks Berordo 3 x 3
Misal $A$ adalah matriks berordo 3 x 3, invers matriks $A$ adalah:
$\begin{array} {lcl} A=\begin{pmatrix} a_{11}\; & a_{12}\; &a_{13} \\ a_{21}\; &a_{22} \; &a_{23} \\ a_{31}\; &a_{32} \; &a_{33} \end{pmatrix}\\\\ Matriks\; kofaktor\;A=\begin{pmatrix} C_{11}\; & C_{12}\; &C_{13} \\ C_{21}\; &C_{22} \; &C_{23} \\ C_{31}\; &C_{32} \; &C_{33} \end{pmatrix} \\\\ Adj\: A=\begin{pmatrix} C_{11}\; & C_{21}\; &C_{31} \\ C_{12}\; &C_{22} \; &C_{32} \\ C_{13}\; &C_{23} \; &C_{33} \end{pmatrix},\; adj\: A\; merupakan\; transpose \; dari\; matriks\; kofaktor \\\\ A^{-1}=\frac{1}{det\: A}\: adj\: A \end{array}$
Contoh:
Carilah invers dari matriks $\begin{array} {lcl} A=\begin{pmatrix} 1\; & 0\; &1 \\ -1\; &3 \; &0 \\ 1\; &0 \; &2 \end{pmatrix} \end{array}$
Pembahasan:
$\begin{array} {lcl} A=\begin{pmatrix} 1\; & 0\; &1 \\ -1\; &3 \; &0 \\ 1\; &0 \; &2 \end{pmatrix}\\ detA=\begin{vmatrix} 1\; & 0\; &1 \\ -1\; &3 \; &0 \\ 1\; &0 \; &2 \end{vmatrix}\begin{matrix} 1\; & 0\\ -1\; &3 \\ 1\; &0 \end{matrix}\\ \; \; \; \; \; \; \; \; =\left ( 1 \right )\left ( 3 \right )\left ( 2 \right )+\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )+\left ( 1 \right )\left ( -1 \right )\left ( 0 \right )-\left ( 1 \right )\left ( 3 \right )\left ( 1 \right )-\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )-\left ( 2 \right )\left ( -1 \right )\left ( 0 \right )\\ \; \; \; \; \; \; \; \; =6+0+0-3-0-0\\ \; \; \; \; \; \; \; \; =3 \end{array}$
$\begin{array} {lcl} C_{11}=\begin{vmatrix} 3\; & 0\\ 0\; &2 \end{vmatrix}=6\\ C_{12}=-\begin{vmatrix} -1\; & 0\\ 1\; &2 \end{vmatrix}=2\\ C_{13}=\begin{vmatrix} -1\; & 3\\ 1\; &0 \end{vmatrix}=-3\\ C_{21}=-\begin{vmatrix} 0\; & 0\\ 0\; &2 \end{vmatrix}=0\\ C_{22}=\begin{vmatrix} 1\; & 1\\ 1\; &2 \end{vmatrix}=1\\ C_{23}=-\begin{vmatrix} 1\; & 0\\ 1\; &0 \end{vmatrix}=0\\ \end{array}$
$\begin{array} {lcl} C_{31}=\begin{vmatrix} 0\; & 1\\ 3\; &0 \end{vmatrix}=-3\\ C_{32}=-\begin{vmatrix} 1\; & 1\\ -1\; &0 \end{vmatrix}=-1\\ C_{33}=\begin{vmatrix} 1\; & 0\\ -1\; &3 \end{vmatrix}=3\\ \end{array}$
$\begin{array} {lcl} Adj\: A=\begin{pmatrix} C_{11}\; & C_{21}\; &C_{31} \\ C_{12}\; &C_{22} \; &C_{32} \\ C_{13}\; &C_{23} \; &C_{33} \end{pmatrix}\\\\ \; \; \; \; \; \; \; \; \; \; =\begin{pmatrix} 6\; & 0\; &-3 \\ 2\; &1 \; &-1 \\ -3\; &0 \; &3 \end{pmatrix}\\\\ A^{-1}=\frac{1}{detA}\, adjA\\ \; \; \; \; \; \; \; =\frac{1}{3}\begin{pmatrix} 6\; & 0\; &-3 \\ 2\; &1 \; &-1 \\ -3\; &0 \; &3 \end{pmatrix}\\ \end{array}$
$\begin{array} {lcl} \; \; \; \; \; \; \; =\begin{pmatrix} 2\; & 0\; &-1 \\ \frac{2}{3}\; &\frac{1}{3} \; &-\frac{1}{3} \\ -1\; &0 \; &1 \end{pmatrix}\\ \end{array}$
Misal $A=\begin{pmatrix} a_{11}\; &a_{12}\; &a_{13} \\ a_{21}\; &a_{22} \; &a_{23} \\ a_{31}\;&a_{32}\; & a_{33} \end{pmatrix}$
$\begin{array}{lcl} detA=|A|=\begin{vmatrix} a_{11}\; &a_{12}\; &a_{13} \\ a_{21}\; &a_{22} \; &a_{23} \\ a_{31}\;&a_{32}\; & a_{33} \end{vmatrix}\begin{matrix} a_{11}\; &a_{12} \\ a_{21}\; & a_{22}\\ a_{31}\; & a_{32} \end{matrix}\\ \; \; \; \; \; \; \; \; =a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{31}a_{22}a_{13}-a_{32}a_{23}a_{11}-a_{33}a_{21}a_{12} \end{array}$
Contoh:
Hitunglah determinan matriks $\begin{array}{lcl} A=\begin{pmatrix} 1\; & 0\; &1 \\ -1\; & 3\; &0 \\ 1\; &0\; &2 \end{pmatrix} \end{array}$
Pembahasan:
$\begin{array}{lcl} detA=\begin{vmatrix} 1\; & 0\; &1 \\ -1\; & 3\; &0 \\ 1\; &0\; &2 \end{vmatrix}\begin{matrix} 1\; &0 \\ -1\; & 3\\ 1\; &0 \end{matrix}\\ \; \; \; \; \; \; \; \; =\left ( 1 \right )\left ( 3 \right )\left ( 2 \right )+\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )+\left ( 1 \right )\left ( -1 \right )\left ( 0 \right )-\left ( 1 \right )\left ( 3 \right )\left ( 1 \right )-\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )-\left ( 2 \right )\left ( -1 \right )\left ( 0 \right )\\ \; \; \; \; \; \; \; \; =6+0+0-3-0-0\\ \; \; \; \; \; \; \; \; =3 \end{array}$
# Determinan Matriks Berordo 3 x 3 Menggunakan Kofaktor
Menentukan minor $(M)$ dan kofaktor $(C)$ dari matriks berordo 3
Misal matriks $\begin{array}{lcl} A=\begin{pmatrix} a_{11}\; &a_{12}\; &a_{13} \\ a_{21}\; &a_{22}\; & a_{23}\\ a_{31}\; &a_{32}\; & a_{33} \end{pmatrix} \end{array}$
Minor dan kofaktor dari matriks $A$
1. $\begin{array}{lcl} M_{11}=\begin{vmatrix} a_{22 }\; & a_{23}\\ a_{32}\; & a_{33} \end{vmatrix}\\ C_{11}=\left ( -1 \right )^{1+1}M_{11}=\begin{vmatrix} a_{22 }\; & a_{23}\\ a_{32}\; & a_{33} \end{vmatrix}\\ \end{array}$
2. $\begin{array}{lcl} M_{12}=\begin{vmatrix} a_{21 }\; & a_{23}\\ a_{31}\; & a_{33} \end{vmatrix}\\ C_{12}=\left ( -1 \right )^{1+2}M_{12}=-\begin{vmatrix} a_{21 }\; & a_{23}\\ a_{31}\; & a_{33} \end{vmatrix}\\ \end{array}$
3. $\begin{array}{lcl} M_{13}=\begin{vmatrix} a_{21 }\; & a_{22}\\ a_{31}\; & a_{32} \end{vmatrix}\\ C_{13}=\left ( -1 \right )^{1+3}M_{13}=\begin{vmatrix} a_{21 }\; & a_{22}\\ a_{31}\; & a_{32} \end{vmatrix}\\ \end{array}$
4. $\begin{array}{lcl} M_{21}=\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{32}\; & a_{33} \end{vmatrix}\\ C_{21}=\left ( -1 \right )^{2+1}M_{21}=-\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{32}\; & a_{33} \end{vmatrix}\\ \end{array}$
5. $\begin{array}{lcl} M_{22}=\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{31}\; & a_{33} \end{vmatrix}\\ C_{22}=\left ( -1 \right )^{2+2}M_{21}=\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{31}\; & a_{33} \end{vmatrix}\\ \end{array}$
6. $\begin{array}{lcl} M_{23}=\begin{vmatrix} a_{11 }\; & a_{12}\\ a_{31}\; & a_{32} \end{vmatrix}\\ C_{23}=\left ( -1 \right )^{2+3}M_{23}=-\begin{vmatrix} a_{11 }\; & a_{12}\\ a_{31}\; & a_{32} \end{vmatrix}\\ \end{array}$
7. $\begin{array}{lcl} M_{31}=\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{22}\; & a_{23} \end{vmatrix}\\ C_{31}=\left ( -1 \right )^{3+1}M_{31}=\begin{vmatrix} a_{12 }\; & a_{13}\\ a_{22}\; & a_{23} \end{vmatrix}\\ \end{array}$
8. $\begin{array}{lcl} M_{32}=\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{21}\; & a_{23} \end{vmatrix}\\ C_{32}=\left ( -1 \right )^{3+2}M_{32}=-\begin{vmatrix} a_{11 }\; & a_{13}\\ a_{21}\; & a_{23} \end{vmatrix}\\ \end{array}$
Determinan dari matriks $A$ (pilih salah satu saja)
$\begin{array}{lcl} 1.\; detA=|A|=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\\ 2.\; detA=|A|=a_{21}C_{21}+a_{22}C_{22}+a_{23}C_{23}\\ 3.\; detA=|A|=a_{31}C_{31}+a_{32}C_{32}+a_{33}C_{33}\\ 4.\; detA=|A|=a_{11}C_{11}+a_{21}C_{21}+a_{31}C_{31}\\ 5.\; detA=|A|=a_{12}C_{12}+a_{22}C_{22}+a_{32}C_{32}\\ 6.\; detA=|A|=a_{13}C_{13}+a_{23}C_{23}+a_{33}C_{33}\\ \ \end{array}$
Contoh:
Hitunglah determinan matriks menggunakan kofaktor $\begin{array}{lcl} A=\begin{pmatrix} 1 \; & 0\; &1 \\ -1\; &3\; &0 \\ 1\; & 0\; &2 \end{pmatrix} \ \end{array}$
Pembahasan:
Untuk menghitung determinan matriks $A$ dengan menggunakan kofaktor,
kita menggunakan rumus 1 (bisa dipilih)
$\begin{array}{lcl} a_{11}=1\\ C_{11}=\begin{vmatrix} 3\; &0 \\ 0\; & 2 \end{vmatrix}\\ C_{11}=6\\\\ a_{12}=0\\ C_{12}=-\begin{vmatrix} -1\; &0 \\ 1\; & 2 \end{vmatrix}\\ C_{12}=2\\ \ \end{array}$
$\begin{array}{lcl} a_{13}=1\\ C_{13}=\begin{vmatrix} -1\; &3 \\ 1\; & 0 \end{vmatrix}\\ C_{13}=0-3\\ C_{13}=-3\\\\ detA=|A|=a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\\ \; \; \; \; \; \; \; \; =\left ( 1 \right )\left ( 6 \right )+\left ( 0 \right )\left ( 2 \right )+\left ( 1 \right )\left ( -3 \right )\\ \; \; \; \; \; \; \; \; =6+0-3\\ \; \; \; \; \; \; \; \; =3 \ \end{array}$
# Invers Matriks Berordo 3 x 3
Misal $A$ adalah matriks berordo 3 x 3, invers matriks $A$ adalah:
$\begin{array} {lcl} A=\begin{pmatrix} a_{11}\; & a_{12}\; &a_{13} \\ a_{21}\; &a_{22} \; &a_{23} \\ a_{31}\; &a_{32} \; &a_{33} \end{pmatrix}\\\\ Matriks\; kofaktor\;A=\begin{pmatrix} C_{11}\; & C_{12}\; &C_{13} \\ C_{21}\; &C_{22} \; &C_{23} \\ C_{31}\; &C_{32} \; &C_{33} \end{pmatrix} \\\\ Adj\: A=\begin{pmatrix} C_{11}\; & C_{21}\; &C_{31} \\ C_{12}\; &C_{22} \; &C_{32} \\ C_{13}\; &C_{23} \; &C_{33} \end{pmatrix},\; adj\: A\; merupakan\; transpose \; dari\; matriks\; kofaktor \\\\ A^{-1}=\frac{1}{det\: A}\: adj\: A \end{array}$
Contoh:
Carilah invers dari matriks $\begin{array} {lcl} A=\begin{pmatrix} 1\; & 0\; &1 \\ -1\; &3 \; &0 \\ 1\; &0 \; &2 \end{pmatrix} \end{array}$
Pembahasan:
$\begin{array} {lcl} A=\begin{pmatrix} 1\; & 0\; &1 \\ -1\; &3 \; &0 \\ 1\; &0 \; &2 \end{pmatrix}\\ detA=\begin{vmatrix} 1\; & 0\; &1 \\ -1\; &3 \; &0 \\ 1\; &0 \; &2 \end{vmatrix}\begin{matrix} 1\; & 0\\ -1\; &3 \\ 1\; &0 \end{matrix}\\ \; \; \; \; \; \; \; \; =\left ( 1 \right )\left ( 3 \right )\left ( 2 \right )+\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )+\left ( 1 \right )\left ( -1 \right )\left ( 0 \right )-\left ( 1 \right )\left ( 3 \right )\left ( 1 \right )-\left ( 0 \right )\left ( 0 \right )\left ( 1 \right )-\left ( 2 \right )\left ( -1 \right )\left ( 0 \right )\\ \; \; \; \; \; \; \; \; =6+0+0-3-0-0\\ \; \; \; \; \; \; \; \; =3 \end{array}$
$\begin{array} {lcl} C_{11}=\begin{vmatrix} 3\; & 0\\ 0\; &2 \end{vmatrix}=6\\ C_{12}=-\begin{vmatrix} -1\; & 0\\ 1\; &2 \end{vmatrix}=2\\ C_{13}=\begin{vmatrix} -1\; & 3\\ 1\; &0 \end{vmatrix}=-3\\ C_{21}=-\begin{vmatrix} 0\; & 0\\ 0\; &2 \end{vmatrix}=0\\ C_{22}=\begin{vmatrix} 1\; & 1\\ 1\; &2 \end{vmatrix}=1\\ C_{23}=-\begin{vmatrix} 1\; & 0\\ 1\; &0 \end{vmatrix}=0\\ \end{array}$
$\begin{array} {lcl} C_{31}=\begin{vmatrix} 0\; & 1\\ 3\; &0 \end{vmatrix}=-3\\ C_{32}=-\begin{vmatrix} 1\; & 1\\ -1\; &0 \end{vmatrix}=-1\\ C_{33}=\begin{vmatrix} 1\; & 0\\ -1\; &3 \end{vmatrix}=3\\ \end{array}$
$\begin{array} {lcl} Adj\: A=\begin{pmatrix} C_{11}\; & C_{21}\; &C_{31} \\ C_{12}\; &C_{22} \; &C_{32} \\ C_{13}\; &C_{23} \; &C_{33} \end{pmatrix}\\\\ \; \; \; \; \; \; \; \; \; \; =\begin{pmatrix} 6\; & 0\; &-3 \\ 2\; &1 \; &-1 \\ -3\; &0 \; &3 \end{pmatrix}\\\\ A^{-1}=\frac{1}{detA}\, adjA\\ \; \; \; \; \; \; \; =\frac{1}{3}\begin{pmatrix} 6\; & 0\; &-3 \\ 2\; &1 \; &-1 \\ -3\; &0 \; &3 \end{pmatrix}\\ \end{array}$
$\begin{array} {lcl} \; \; \; \; \; \; \; =\begin{pmatrix} 2\; & 0\; &-1 \\ \frac{2}{3}\; &\frac{1}{3} \; &-\frac{1}{3} \\ -1\; &0 \; &1 \end{pmatrix}\\ \end{array}$
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di c21 sprtinya trdapat ksalahan
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