Math Problem Statement
Find A^2 + A + I for the given matrix A where A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]].
Solution
The matrix in the image is given as:
1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}$$ You are asked to compute $$A^2 + A + I$$, where $$A^2$$ is the square of the matrix, $$A$$ is the original matrix, and $$I$$ is the identity matrix of the same size. ### Step-by-step Solution: 1. **Matrix $$A$$** is already given: $$A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}$$ 2. **Find $$A^2$$ (matrix multiplication):** We compute the product of $$A \cdot A$$: $$A^2 = A \cdot A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix}$$ We will multiply the matrices now. $$A^2 = \begin{pmatrix} (1 \times 1) + (2 \times 2) + (2 \times 2) & (1 \times 2) + (2 \times 1) + (2 \times 2) & (1 \times 2) + (2 \times 2) + (2 \times 1) \\ (2 \times 1) + (1 \times 2) + (2 \times 2) & (2 \times 2) + (1 \times 1) + (2 \times 2) & (2 \times 2) + (1 \times 2) + (2 \times 1) \\ (2 \times 1) + (2 \times 2) + (1 \times 2) & (2 \times 2) + (2 \times 1) + (1 \times 2) & (2 \times 2) + (2 \times 2) + (1 \times 1) \end{pmatrix}$$ Simplifying: $$A^2 = \begin{pmatrix} 1 + 4 + 4 & 2 + 2 + 4 & 2 + 4 + 2 \\ 2 + 2 + 4 & 4 + 1 + 4 & 4 + 2 + 2 \\ 2 + 4 + 2 & 4 + 2 + 2 & 4 + 4 + 1 \end{pmatrix} = \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix}$$ So, $$A^2 = \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix}$$. 3. **Identity matrix $$I$$** (since $$A$$ is 3x3, we use a 3x3 identity matrix): $$I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ 4. **Add $$A^2$$, $$A$$, and $$I$$:** $$A^2 + A + I = \begin{pmatrix} 9 & 8 & 8 \\ 8 & 9 & 8 \\ 8 & 8 & 9 \end{pmatrix} + \begin{pmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ First, sum $$A^2 + A$$: $$A^2 + A = \begin{pmatrix} 9+1 & 8+2 & 8+2 \\ 8+2 & 9+1 & 8+2 \\ 8+2 & 8+2 & 9+1 \end{pmatrix} = \begin{pmatrix} 10 & 10 & 10 \\ 10 & 10 & 10 \\ 10 & 10 & 10 \end{pmatrix}$$ Then, sum the result with $$I$$: $$A^2 + A + I = \begin{pmatrix} 10+1 & 10+0 & 10+0 \\ 10+0 & 10+1 & 10+0 \\ 10+0 & 10+0 & 10+1 \end{pmatrix} = \begin{pmatrix} 11 & 10 & 10 \\ 10 & 11 & 10 \\ 10 & 10 & 11 \end{pmatrix}$$ ### Final result: $$A^2 + A + I = \begin{pmatrix} 11 & 10 & 10 \\ 10 & 11 & 10 \\ 10 & 10 & 11 \end{pmatrix}$$ Would you like a further breakdown or have any other questions? ### Relative questions: 1. How is matrix multiplication performed? 2. What is the significance of the identity matrix in matrix algebra? 3. Can we diagonalize the given matrix $$A$$? 4. How do eigenvalues relate to matrix powers? 5. What properties does the resulting matrix exhibit? ### Tip: When performing matrix operations, double-check each row and column operation to avoid mistakes in addition and multiplication steps.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Operations
Matrix Multiplication
Matrix Addition
Identity Matrix
Formulas
Matrix Multiplication Formula: C = AB
Identity Matrix Addition: A + I
Theorems
Properties of Identity Matrix
Matrix Multiplication Associativity
Suitable Grade Level
Grades 10-12