Math Problem Statement
Find A^2 + 5A + 7I for the matrix A = [[3, 1, 4], [2, 1, 3], [2, 2, 1]].
Solution
The given problem involves a matrix and the task is to find the expression , where is the identity matrix.
Matrix :
3 & 1 & 4 \\ 2 & 1 & 3 \\ 2 & 2 & 1 \end{pmatrix}$$ ### Steps to solve $$A^2 + 5A + 7I$$: 1. **Compute $$A^2$$ (matrix multiplication of $$A$$ with itself):** $$A^2 = A \times A$$ 2. **Multiply $$A$$ by 5:** $$5A = 5 \times A$$ 3. **Multiply the identity matrix $$I$$ by 7:** $$7I = 7 \times I$$ where $$I$$ is the 3x3 identity matrix: $$I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ 4. **Add the resulting matrices from steps 1, 2, and 3 to get the final result:** $$A^2 + 5A + 7I$$ Let me calculate this for you.The result of the expression $$A^2 + 5A + 7I$$ is: $$A^2 + 5A + 7I = \begin{pmatrix} 41 & 17 & 39 \\ 24 & 21 & 29 \\ 22 & 16 & 27 \end{pmatrix}$$ Would you like a detailed explanation of the steps? Here are some related questions to further explore: 1. How do you perform matrix multiplication? 2. What are the properties of identity matrices in operations like this? 3. Can matrix powers be applied to non-square matrices? 4. How do scalar multiplications of matrices affect their elements? 5. How does matrix addition work element-wise? **Tip:** When multiplying matrices, ensure the number of columns in the first matrix matches the number of rows in the second for a valid multiplication.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Matrix Addition
Scalar Multiplication
Identity Matrix
Formulas
A^2 + 5A + 7I
Matrix multiplication formula: (AB)_ij = sum(A_ik * B_kj)
Identity matrix: I = diag(1,1,1)
Theorems
Properties of Matrix Multiplication
Distributive Property of Matrices
Identity Matrix in Linear Algebra
Suitable Grade Level
Undergraduate Level