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Question Analysis and Solution:

(a) Tentukan sama ada dua matriks berikut SAMA atau TIDAK

1. (i) Matriks pertama: 1 & -4 & 5 \\ 2 & 8 & 7 \end{bmatrix}$$ **dan matriks kedua:** $$\begin{bmatrix} 1 & -4 & 5 \\ 2 & 8 & 7 \end{bmatrix}$$ These matrices are identical. **Answer:** Sama.
2. (ii) Matriks pertama: 7 & 2 \\ 1 & 5 \end{bmatrix}$$ **dan matriks kedua:** $$\begin{bmatrix} 3.5 & 3 \\ 0.2 & 4 \end{bmatrix}$$ These matrices are not identical. **Answer:** Tidak sama.

(b) Diberi matriks $A$ dan $B$, cari:

1. $B^T$: Transpose of $B$: 3 & 2 \\ 1 & 0 \end{bmatrix}$$ $$B^T = \begin{bmatrix} 3 & 1 \\ 2 & 0 \end{bmatrix}$$
2. $A^{-1}$: Given: 3 & -4 \\ 1 & 2 \end{bmatrix}$$ Calculate determinant: $$\text{det}(A) = (3)(2) - (1)(-4) = 6 + 4 = 10$$ Inverse: $$A^{-1} = \frac{1}{\text{det}(A)} \begin{bmatrix} 2 & 4 \\ -1 & 3 \end{bmatrix} = \frac{1}{10} \begin{bmatrix} 2 & 4 \\ -1 & 3 \end{bmatrix} = \begin{bmatrix} 0.2 & 0.4 \\ -0.1 & 0.3 \end{bmatrix}$$
3. $A^T + 3B$: Transpose of $A$: 3 & 1 \\ -4 & 2 \end{bmatrix}$$ Multiply $$B$$ by 3: $$3B = 3 \cdot \begin{bmatrix} 3 & 2 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} 9 & 6 \\ 3 & 0 \end{bmatrix}$$ Add: $$A^T + 3B = \begin{bmatrix} 3 & 1 \\ -4 & 2 \end{bmatrix} + \begin{bmatrix} 9 & 6 \\ 3 & 0 \end{bmatrix} = \begin{bmatrix} 12 & 7 \\ -1 & 2 \end{bmatrix}$$

(c) Cari nilai $r$:

$r \cdot 1 + 1 \cdot 4 = 7$ $r + 4 = 7$ $r = 3$

Summary of Answers:

(a) (i) Sama, (ii) Tidak sama
(b) (i) $B^T = \begin{bmatrix} 3 & 1 \\ 2 & 0 \end{bmatrix}$,
(ii) $A^{-1} = \begin{bmatrix} 0.2 & 0.4 \\ -0.1 & 0.3 \end{bmatrix}$,
(iii) $A^T + 3B = \begin{bmatrix} 12 & 7 \\ -1 & 2 \end{bmatrix}$
(c) $r = 3$

Would you like further explanations or details on any of these?

Related Questions:

1. How do you calculate the inverse of a 2x2 matrix systematically?
2. What are the properties of matrix transpose operations?
3. How does scalar multiplication of a matrix affect its determinant?
4. Why is it important to check the determinant before finding the inverse of a matrix?
5. How do you solve linear equations using matrices?

Tip:

When finding the inverse of a matrix, always ensure the determinant is non-zero; otherwise, the matrix is not invertible.