Math Problem Statement
Tentukan nilai variabel a, b, dan c pada operasi matriks berikut.
Solution
The problem you've shared involves solving for the variables , , and in a system of matrix equations. Let's break it down step by step.
For part (a):
The equation given is:
-1 & 0 \\ 3 & 2 \\ 5 & 2 \end{pmatrix} - 3 \cdot \begin{pmatrix} 0 & -1 \\ 5 & 4 \\ -3 & 4 \end{pmatrix} = \begin{pmatrix} a & 3 \\ -12 & 2b \\ -7c & -10 \end{pmatrix}$$ We first calculate the left-hand side. Multiply the second matrix by 3: $$3 \cdot \begin{pmatrix} 0 & -1 \\ 5 & 4 \\ -3 & 4 \end{pmatrix} = \begin{pmatrix} 0 & -3 \\ 15 & 12 \\ -9 & 12 \end{pmatrix}$$ Now subtract: $$\begin{pmatrix} -1 & 0 \\ 3 & 2 \\ 5 & 2 \end{pmatrix} - \begin{pmatrix} 0 & -3 \\ 15 & 12 \\ -9 & 12 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ 3 - 15 & 2 - 12 \\ 5 + 9 & 2 - 12 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ -12 & -10 \\ 14 & -10 \end{pmatrix}$$ Equating the matrices: $$\begin{pmatrix} a & 3 \\ -12 & 2b \\ -7c & -10 \end{pmatrix} = \begin{pmatrix} -1 & 3 \\ -12 & -10 \\ 14 & -10 \end{pmatrix}$$ From this, we find: 1. $$a = -1$$ 2. $$2b = -10 \implies b = -5$$ 3. $$-7c = 14 \implies c = -2$$ --- ### For part (b): The equation given is: $$\begin{pmatrix} -1 & 5 & 2 \\ 3 & 2 & 4 \\ -2 & 1 & 0 \end{pmatrix} + 2 \cdot \begin{pmatrix} 1 & a & 5 \\ -4 & -2 & 2 \\ a + b & 3 - 2c & -1 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 12 \\ -5 & -2 & 8 \\ -4 & 7 & 3b \end{pmatrix}$$ First, multiply the second matrix byAsk a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Scalar Multiplication
Matrix Addition
Matrix Equality
Formulas
Matrix Subtraction: A - B = C, where A and B are matrices
Scalar Multiplication: k * A, where k is a scalar and A is a matrix
Matrix Equality: If two matrices are equal, their corresponding elements are equal
Theorems
Matrix Equality Theorem
Suitable Grade Level
Undergraduate or Senior High School