Jawaban Soal Latihan 1 Matriks

Soal Latihan 1

1. Diketahui matriks A sebagai berikut :
   
   A = $\left(\left(,,\begin{array}{ccccc}9& 0& 3& -7& 6\\ -5& 6& 8& 1& 4\\ 5& 7& 0& 2& 8\end{array},,\right)\right)$
   
   
   1. Sebutkan ordo dari matriks A
   2. Sebutkan elemen-elemen kolom ke 2
   3. Sebutkan elemen-elemen baris ke 3
   4. Jika a_ij menyatakan elemen baris ke i dan kolom ke j maka tentukan a₂₃, a₁₄ dan a₃₅


2. Berikan contoh matriks berordo 3x3 untuk :
   1. Matriks diagonal
   2. Matriks identitas
   3. Matriks segitiga atas
   4. Matriks segitiga bawah


3. Tulislah transpose dari matriks berikut :
   
   
   1. A = $\left(\left(,,\begin{array}{cc}4& 3\\ 5& 6\end{array},,\right)\right)$
   
   
   2. B = $\left(\left(,,\begin{array}{cc}2& 4\\ -3& 1\\ 0& 5\end{array},,\right)\right)$
   
   
   3. C = $\left(\left(,,\begin{array}{ccc}4& 5& -1\\ 2& 3& 0\end{array},,\right)\right)$
   
   
   4. D = $\left(\left(,,\begin{array}{ccc}4& 0& 5\\ 2& 4& 7\\ -1& 3& -2\end{array},,\right)\right)$


4. Tentukan nilai x dan y dari persamaan berikut :
   
   
   1. $\left(\left(,,\begin{array}{cc}\mathrm{x}& \mathrm{2y}\\ 0& 3\end{array},,\right)\right)=\left(\left(,,\begin{array}{cc}1& 8\\ 0& 3\end{array},,\right)\right)$
   
   
   2. $\left(\left(,,\begin{array}{cc}\mathrm{x\; +\; y}& 5\\ 4& \mathrm{x\; -\; y}\end{array},,\right)\right)=\left(\left(,,\begin{array}{cc}5& 5\\ 4& 1\end{array},,\right)\right)$
   
   
   3. $\left(\left(,,\begin{array}{cc}\mathrm{x}& 4\\ \mathrm{2y}& \mathrm{3z}\end{array},,\right)\right)=\left(\left(,,\begin{array}{cc}\mathrm{4z\; -\; 6y}& \mathrm{2x}\\ \mathrm{4x\; +\; 2}& \mathrm{2y\; +\; 14}\end{array},,\right)\right)$


5. Tentukan x dan y jika A = B^t
   
   
   1. A = $\left(\left(,,\begin{array}{cc}\mathrm{2x}& 0\\ 1& \mathrm{-4y}\end{array},,\right)\right)$ dan B = $\left(\left(,,\begin{array}{cc}8& 1\\ 0& 40\end{array},,\right)\right)$
   
   
   2. A = $\left(\left(,,\begin{array}{cc}\mathrm{x\; -\; 2y}& 1\\ 0& \mathrm{-3y}\end{array},,\right)\right)$ dan B = $\left(\left(,,\begin{array}{cc}8& 0\\ 1& 40\end{array},,\right)\right)$

Jawab :

1. 1. ordo mariks A = 3 x 5
   2. elemen-elemen kolom ke 2 = 0, 6, 7
   3. elemen-elemen baris ke 3 = 5, 7, 0, 2, 8
   4. a₂₃ = 8
      a₁₄ = -7
      a₃₅ = 8


2. 1. matriks diagonal = $\left(\left(,,\begin{array}{ccc}1& 0& 0\\ 0& 2& 0\\ 0& 0& 3\end{array},,\right)\right)$
   
   
   2. matriks identitas = $\left(\left(,,\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array},,\right)\right)$
   
   
   3. matriks segitiga atas = $\left(\left(,,\begin{array}{ccc}4& 1& 5\\ 0& 2& 7\\ 0& 0& 3\end{array},,\right)\right)$
   
   
   4. matriks segitiga bawah = $\left(\left(,,\begin{array}{ccc}4& 0& 0\\ 2& 6& 0\\ 5& 3& 1\end{array},,\right)\right)$


3. 1. A^t = $\left(\left(,,\begin{array}{cc}4& 5\\ 3& 6\end{array},,\right)\right)$
   
   
   2. B^t = $\left(\left(,,\begin{array}{ccc}2& -3& 0\\ 4& 1& 5\end{array},,\right)\right)$
   
   
   3. C^t = $\left(\left(,,\begin{array}{cc}4& 2\\ 5& 3\\ -1& 0\end{array},,\right)\right)$
   
   
   4. D^t = $\left(\left(,,\begin{array}{ccc}4& 2& -1\\ 0& 4& 3\\ 5& 7& -2\end{array},,\right)\right)$


4. 1. x = 1    2y = 8
             y = 4
   
   
   2. x + y = 5
      x = 5 - y
      
      x - y = 1
      (5 - y) - y = 1
      5 - 2y = 1
      -2y = 1 - 5
      -2y = -4
      y = 2
      
      x = 5 - y
      x = 5 - 2
      x = 3
   
   
   3. 4 = 2x
      x = 2
      
      2y = 4x + 2
      2y = 4(2) + 2
      2y = 8 + 2
      2y = 10
      y = 5
      
      3z = 2y + 14
      3z = 2(5) + 14
      3z = 10 + 14
      3z = 24
      z = 8


5. 1. B^t = $\left(\left(,,\begin{array}{cc}8& 0\\ 1& 40\end{array},,\right)\right)$
      
      A = B^t
      $\left(\left(,,\begin{array}{cc}\mathrm{2x}& 0\\ 1& \mathrm{-4y}\end{array},,\right)\right)$ = $\left(\left(,,\begin{array}{cc}8& 0\\ 1& 40\end{array},,\right)\right)$
      
      2x = 8
      x = 4
      
      -4y = 40
      y = -10
   
   
   2. B^t = $\left(\left(,,\begin{array}{cc}8& 1\\ 0& 40\end{array},,\right)\right)$
      
      A = B^t
      $\left(\left(,,\begin{array}{cc}\mathrm{x\; -\; 2y}& 1\\ 0& \mathrm{-3y}\end{array},,\right)\right)$ = $\left(\left(,,\begin{array}{cc}8& 1\\ 0& 40\end{array},,\right)\right)$
      
      -3y = 40
      $\mathrm{y}=-\frac{40}{3}$
      
      x - 2y = 8
      $\mathrm{x}-2(-\frac{40}{3})=8$
      $\mathrm{x}+\frac{80}{3}=8$
      $\mathrm{x}=8-\frac{80}{3}$
      $\mathrm{x}=\frac{24}{3}-\frac{80}{3}$
      $\mathrm{x}=-\frac{56}{3}$