Math Problem Statement

Solve for X in the equation X + 3A = B given A = [[-2, -1], [1, 0], [3, -4]] and B = [[0, 3], [2, 0], [-4, -1]].

Solution

We are given the equation X+3A=BX + 3A = B, where:

A=[211034],B=[032041]A = \begin{bmatrix} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{bmatrix}, \quad B = \begin{bmatrix} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{bmatrix}

To solve for XX, we can rearrange the equation to isolate XX:

X=B3AX = B - 3A

Now, let's calculate 3A3A:

3A=3×[211034]=[6330912]3A = 3 \times \begin{bmatrix} -2 & -1 \\ 1 & 0 \\ 3 & -4 \end{bmatrix} = \begin{bmatrix} -6 & -3 \\ 3 & 0 \\ 9 & -12 \end{bmatrix}

Next, subtract 3A3A from BB:

X=[032041][6330912]X = \begin{bmatrix} 0 & 3 \\ 2 & 0 \\ -4 & -1 \end{bmatrix} - \begin{bmatrix} -6 & -3 \\ 3 & 0 \\ 9 & -12 \end{bmatrix}

Perform the subtraction element-wise:

= \begin{bmatrix} 6 & 6 \\ -1 & 0 \\ -13 & 11 \end{bmatrix}$$ Thus, the solution is: $$X = \begin{bmatrix} 6 & 6 \\ -1 & 0 \\ -13 & 11 \end{bmatrix}$$ This corresponds to the fourth option in the multiple-choice answers. Would you like any additional details or explanations? ### Related Questions: 1. How do you multiply a matrix by a scalar? 2. Can you explain matrix subtraction in more detail? 3. What properties do matrix equations like $$X + 3A = B$$ follow? 4. How can we check if the solution for $$X$$ is correct? 5. What happens if $$A$$ or $$B$$ are not the same dimensions? ### Tip: When solving matrix equations, always ensure the dimensions of the matrices match for addition and subtraction.

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Math Problem Analysis

Mathematical Concepts

Matrix Operations
Matrix Addition
Matrix Multiplication by a Scalar
Matrix Subtraction

Formulas

Matrix equation: X + 3A = B
Scalar multiplication: cA = c * A
Matrix subtraction: X = B - cA

Theorems

Properties of Matrix Addition
Properties of Scalar Multiplication

Suitable Grade Level

Grades 10-12