Math Problem Statement
A=2 -1 3 0 et B=5 4 1 8 , Calculer (A+B)².
Solution
To calculate , let's proceed step by step:
Given Matrices:
\begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 4 \\ 1 & 8 \end{bmatrix}.$$ ### Step 1: Compute $$A + B$$ $$A + B = \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix} + \begin{bmatrix} 5 & 4 \\ 1 & 8 \end{bmatrix} = \begin{bmatrix} 2+5 & -1+4 \\ 3+1 & 0+8 \end{bmatrix} = \begin{bmatrix} 7 & 3 \\ 4 & 8 \end{bmatrix}.$$ ### Step 2: Compute $$(A + B)^2 = (A + B) \cdot (A + B)$$ $$(A + B)^2 = \begin{bmatrix} 7 & 3 \\ 4 & 8 \end{bmatrix} \cdot \begin{bmatrix} 7 & 3 \\ 4 & 8 \end{bmatrix}.$$ Performing matrix multiplication: $$\begin{aligned} & \begin{bmatrix} 7 & 3 \\ 4 & 8 \end{bmatrix} \cdot \begin{bmatrix} 7 & 3 \\ 4 & 8 \end{bmatrix} = \begin{bmatrix} (7 \cdot 7 + 3 \cdot 4) & (7 \cdot 3 + 3 \cdot 8) \\ (4 \cdot 7 + 8 \cdot 4) & (4 \cdot 3 + 8 \cdot 8) \end{bmatrix} \\[10pt] & = \begin{bmatrix} 49 + 12 & 21 + 24 \\ 28 + 32 & 12 + 64 \end{bmatrix} = \begin{bmatrix} 61 & 45 \\ 60 & 76 \end{bmatrix}. \end{aligned}$$ ### Final Answer: $$(A + B)^2 = \begin{bmatrix} 61 & 45 \\ 60 & 76 \end{bmatrix}.$$ Let me know if you have further questions or want a detailed explanation! 😊 --- **Follow-Up Questions for Exploration:** 1. How do we compute matrix multiplication in general for larger matrices? 2. What is the significance of the square of a matrix in linear algebra? 3. Can a non-square matrix have a square operation defined? 4. How would the result differ if we were calculating $$(A + B)^3$$? 5. Can this computation be simplified using properties of symmetric matrices if $$A$$ and $$B$$ had symmetry? **Tip:** When multiplying matrices, always confirm that the number of columns in the first matrix equals the number of rows in the second matrix!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Addition
Matrix Multiplication
Exponents of Matrices
Formulas
(A + B) = A + B (element-wise addition)
(A + B)^2 = (A + B) × (A + B)
Theorems
Matrix Multiplication Rules
Suitable Grade Level
Grades 10-12