Math Problem Statement
Diberi matriks A = [[-1, 0], [3, 5]], B = [[2, 0], [4, -3], [0, -1]], dan C = [6, -2]. Kirakan: (i) (1/2)C, (ii) CA, (iii) A^2, (iv) 2BA.
Solution
Let's solve the given problem step by step.
Problem Overview:
We have matrices:
B = \begin{pmatrix} 2 & 0 \\ 4 & -3 \\ 0 & -1 \end{pmatrix}, C = \begin{pmatrix} 6 \\ -2 \end{pmatrix}.$$ We need to solve the following: 1. $$\frac{1}{2}C$$ 2. $$CA$$ 3. $$A^2$$ 4. $$2BA$$ --- ### Step 1: $$\frac{1}{2}C$$ $$\frac{1}{2}C = \frac{1}{2} \begin{pmatrix} 6 \\ -2 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}.$$ --- ### Step 2: $$CA$$ Since $$C$$ is $$2 \times 1$$ and $$A$$ is $$2 \times 2$$, the multiplication $$CA$$ is **not defined** because the number of columns in $$C$$ does not match the number of rows in $$A$$. --- ### Step 3: $$A^2$$ $$A^2 = A \cdot A = \begin{pmatrix} -1 & 0 \\ 3 & 5 \end{pmatrix} \cdot \begin{pmatrix} -1 & 0 \\ 3 & 5 \end{pmatrix}.$$ Performing the matrix multiplication: $$A^2 = \begin{pmatrix} (-1)(-1) + (0)(3) & (-1)(0) + (0)(5) \\ (3)(-1) + (5)(3) & (3)(0) + (5)(5) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 12 & 25 \end{pmatrix}.$$ --- ### Step 4: $$2BA$$ $$2BA = 2 \cdot \left(B \cdot A\right).$$ First, calculate $$B \cdot A$$: $$B \cdot A = \begin{pmatrix} 2 & 0 \\ 4 & -3 \\ 0 & -1 \end{pmatrix} \cdot \begin{pmatrix} -1 & 0 \\ 3 & 5 \end{pmatrix}.$$ Performing the multiplication: $$B \cdot A = \begin{pmatrix} (2)(-1) + (0)(3) & (2)(0) + (0)(5) \\ (4)(-1) + (-3)(3) & (4)(0) + (-3)(5) \\ (0)(-1) + (-1)(3) & (0)(0) + (-1)(5) \end{pmatrix} = \begin{pmatrix} -2 & 0 \\ -13 & -15 \\ -3 & -5 \end{pmatrix}.$$ Now multiply by 2: $$2BA = 2 \cdot \begin{pmatrix} -2 & 0 \\ -13 & -15 \\ -3 & -5 \end{pmatrix} = \begin{pmatrix} -4 & 0 \\ -26 & -30 \\ -6 & -10 \end{pmatrix}.$$ --- ### Final Answers: 1. $$\frac{1}{2}C = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$$ 2. $$CA$$: **Not defined** 3. $$A^2 = \begin{pmatrix} 1 & 0 \\ 12 & 25 \end{pmatrix}$$ 4. $$2BA = \begin{pmatrix} -4 & 0 \\ -26 & -30 \\ -6 & -10 \end{pmatrix}$$ Let me know if you want detailed steps for each calculation! --- #### Further Questions: 1. How can matrix multiplication rules help identify undefined products quickly? 2. What are the implications of squaring a matrix, and why does $$A^2$$ result in this specific pattern? 3. Could $$CA$$ be redefined if $$C$$ were reshaped? How? 4. How is scalar multiplication applied uniformly across matrices like in $$2BA$$? 5. What happens when you try to compute higher powers of matrix $$A$$, like $$A^3$$? #### Tip: Always check matrix dimensions before performing multiplication to avoid errors in computation!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Scalar Multiplication
Matrix Squaring
Formulas
Matrix multiplication: (AB)_{ij} = sum(A_{ik} * B_{kj})
Scalar multiplication: k * A = [k * a_ij]
Matrix squaring: A^2 = A * A
Theorems
Matrix Multiplication Dimension Rule
Properties of Matrix Squaring
Suitable Grade Level
Grades 10-12