Math Problem Statement
What is A^2 for the given matrix A?
Solution
From the image, it appears you are dealing with a matrix and you are calculating .
Here are the steps to calculate :
Step 1: Define the matrix
The given matrix seems to be:
\begin{bmatrix} 2 & -1 & 1 \\ -1 & 3 & -3 \\ 2 & -1 & -1 \end{bmatrix}$$ ### Step 2: Compute $$A^2 = A \cdot A$$ Perform matrix multiplication $$A \cdot A$$ by calculating each entry $$(i, j)$$ in the resulting matrix as: $$(A^2)_{i,j} = \sum_{k=1}^{n} A_{i,k} \cdot A_{k,j},$$ where $$n$$ is the number of rows/columns (since $$A$$ is a square matrix). ### Step 3: Perform the multiplication Let me calculate this explicitly. import numpy as np # Define the matrix A A = np.array([ [2, -1, 1], [-1, 3, -3], [2, -1, -1] ]) # Compute A^2 A_squared = np.dot(A, A) # Display the result A_squaredarray([[ 7, -6, 4], [-11, 13, -7], [ 3, -4, 6]])array([[ 7, -6, 4], [-11, 13, -7], [ 3, -4, 6]])successThe result of $$A^2 = A \cdot A$$ is: $$A^2 = \begin{bmatrix} 7 & -6 & 4 \\ -11 & 13 & -7 \\ 3 & -4 & 6 \end{bmatrix}$$ Let me know if you'd like a detailed step-by-step explanation of the calculation or if you have further questions! Here are 5 related questions for further exploration: 1. What are the eigenvalues and eigenvectors of the matrix $$A$$? 2. Is the matrix $$A$$ invertible? If so, calculate $$A^{-1}$$. 3. What is the determinant of $$A$$? 4. How does $$A^3$$ look, and how does it relate to $$A^2$$? 5. Can we diagonalize $$A$$, and what would the diagonal matrix look like? **Tip:** Matrix multiplication is associative but not commutative, so always maintain the correct order when multiplying matrices!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Formulas
(A^2)_{i,j} = Σ A_{i,k} * A_{k,j}
Matrix multiplication rules
Theorems
Associativity of Matrix Multiplication
Suitable Grade Level
Undergraduate (Linear Algebra Course)