Math Problem Statement
If A = [[-1, 3, 3], [0, 1, 3], [-2, 3, -2]], then find A^2.
Solution
To solve for , we need to multiply the given matrix by itself. The matrix is:
-1 & 3 & 3 \\ 0 & 1 & 3 \\ -2 & 3 & -2 \end{pmatrix}$$ We can calculate $$A^2 = A \times A$$ by performing matrix multiplication. I will walk you through this step-by-step. ### Matrix Multiplication: Each element in the product $$A^2$$ is calculated as the dot product of a row from the first matrix with a column from the second matrix. #### First row, first column: $$(-1 \times -1) + (3 \times 0) + (3 \times -2) = 1 + 0 - 6 = -5$$ #### First row, second column: $$(-1 \times 3) + (3 \times 1) + (3 \times 3) = -3 + 3 + 9 = 9$$ #### First row, third column: $$(-1 \times 3) + (3 \times 3) + (3 \times -2) = -3 + 9 - 6 = 0$$ #### Second row, first column: $$(0 \times -1) + (1 \times 0) + (3 \times -2) = 0 + 0 - 6 = -6$$ #### Second row, second column: $$(0 \times 3) + (1 \times 1) + (3 \times 3) = 0 + 1 + 9 = 10$$ #### Second row, third column: $$(0 \times 3) + (1 \times 3) + (3 \times -2) = 0 + 3 - 6 = -3$$ #### Third row, first column: $$(-2 \times -1) + (3 \times 0) + (-2 \times -2) = 2 + 0 + 4 = 6$$ #### Third row, second column: $$(-2 \times 3) + (3 \times 1) + (-2 \times 3) = -6 + 3 - 6 = -9$$ #### Third row, third column: $$(-2 \times 3) + (3 \times 3) + (-2 \times -2) = -6 + 9 + 4 = 7$$ ### Final Result: $$A^2 = \begin{pmatrix} -5 & 9 & 0 \\ -6 & 10 & -3 \\ 6 & -9 & 7 \end{pmatrix}$$ Would you like me to go over any specific details, or do you have questions about the calculation? Here are 5 follow-up questions for deeper understanding: 1. What are the steps involved in performing matrix multiplication in general? 2. How can you verify if your matrix multiplication result is correct? 3. Can matrix multiplication be performed in a different order (i.e., is it commutative)? 4. What are the properties of square matrices when multiplied by themselves (powers of matrices)? 5. How would the result differ if matrix $$A$$ were diagonal or triangular? **Tip**: When multiplying matrices, remember that the order of multiplication matters. In general, $$A \times B \neq B \times A$$.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Algebra
Formulas
Matrix multiplication: (AB)_{ij} = Σ (A_{ik} * B_{kj})
Theorems
Properties of matrix multiplication
Matrix powers
Suitable Grade Level
Undergraduate (Linear Algebra)