Math Problem Statement
Solution
The given exercise asks for operations involving the matrices , , , and . I'll solve each part step-by-step. The matrices provided are:
B = \begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix}, \quad C = \begin{bmatrix} -1 & 2 \\ 2 & 4 \end{bmatrix}, \quad D = \begin{bmatrix} 2 \\ -1 \end{bmatrix}.$$ ### (a) $$A + B$$ Add corresponding elements of $$A$$ and $$B$$: $$A + B = \begin{bmatrix} 1 & 2 \\ 1 & -1 \end{bmatrix} + \begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} -1 & 2 \\ 4 & 0 \end{bmatrix}.$$ ### (b) $$A \cdot C$$ Matrix multiplication of $$A$$ and $$C$$: $$A \cdot C = \begin{bmatrix} 1 & 2 \\ 1 & -1 \end{bmatrix} \cdot \begin{bmatrix} -1 & 2 \\ 2 & 4 \end{bmatrix}.$$ Perform the multiplication: $$A \cdot C = \begin{bmatrix} (1)(-1) + (2)(2) & (1)(2) + (2)(4) \\ (1)(-1) + (-1)(2) & (1)(2) + (-1)(4) \end{bmatrix} = \begin{bmatrix} 3 & 10 \\ -3 & -2 \end{bmatrix}.$$ ### (c) $$B \cdot C$$ Matrix multiplication of $$B$$ and $$C$$: $$B \cdot C = \begin{bmatrix} -2 & 0 \\ 3 & 1 \end{bmatrix} \cdot \begin{bmatrix} -1 & 2 \\ 2 & 4 \end{bmatrix}.$$ Perform the multiplication: $$B \cdot C = \begin{bmatrix} (-2)(-1) + (0)(2) & (-2)(2) + (0)(4) \\ (3)(-1) + (1)(2) & (3)(2) + (1)(4) \end{bmatrix} = \begin{bmatrix} 2 & -4 \\ -1 & 10 \end{bmatrix}.$$ ### (d) $$C \cdot D$$ Matrix multiplication of $$C$$ and $$D$$: $$C \cdot D = \begin{bmatrix} -1 & 2 \\ 2 & 4 \end{bmatrix} \cdot \begin{bmatrix} 2 \\ -1 \end{bmatrix}.$$ Perform the multiplication: $$C \cdot D = \begin{bmatrix} (-1)(2) + (2)(-1) \\ (2)(2) + (4)(-1) \end{bmatrix} = \begin{bmatrix} -4 \\ 0 \end{bmatrix}.$$ ### (e) $$D \cdot A$$ Matrix multiplication of $$D$$ and $$A$$ (dimensions do not align for this multiplication). **Not possible**. ### (f) $$D \cdot B$$ Matrix multiplication of $$D$$ and $$B$$ (dimensions do not align for this multiplication). **Not possible**. ### (g) $$-A$$ Negate all elements of $$A$$: $$-A = \begin{bmatrix} -1 & -2 \\ -1 & 1 \end{bmatrix}.$$ ### (h) $$-D$$ Negate all elements of $$D$$: $$-D = \begin{bmatrix} -2 \\ 1 \end{bmatrix}.$$ Let me know if you need detailed explanations of these calculations, or further assistance! --- **Relative Questions:** 1. Can you calculate the determinant of $$A$$ or $$B$$? 2. How do you interpret the resulting matrices geometrically? 3. What are the eigenvalues and eigenvectors of $$C$$? 4. Can you verify if $$C$$ is invertible? 5. How would you interpret the product $$A \cdot C$$ in a real-world context? **Tip:** Always ensure matrix dimensions are compatible before performing multiplication.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Matrix Addition
Matrix Multiplication
Negation of Matrices
Formulas
Matrix Addition: A + B = [a11 + b11, a12 + b12; a21 + b21, a22 + b22]
Matrix Multiplication: A * B = [a11*b11 + a12*b21, a11*b12 + a12*b22; a21*b11 + a22*b21, a21*b12 + a22*b22]
Negation of Matrix: -A = [-a11, -a12; -a21, -a22]
Theorems
Properties of Matrix Addition
Matrix Multiplication Rules
Negation of Matrix Elements
Suitable Grade Level
Grades 10-12
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