Math Problem Statement
Let A = ( 1 3 4 1 3 2 4 10 9 ) , B = ( 7 13 -6 -1-7 2-2 2 0 ) , C = ( 1 1 2 ) ii) evaluate ( A^2 - B transpose ) C
Solution
We are tasked with evaluating , where , , and are defined as:
B = \begin{pmatrix} 7 & 13 & -6 \\ -1 & -7 & 2 \\ -2 & 2 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}$$ ### Step 1: Compute $$A^2$$ To calculate $$A^2$$, we multiply $$A$$ by itself: $$A^2 = A \times A = \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix} \times \begin{pmatrix} 1 & 3 & 4 \\ 1 & 3 & 2 \\ 4 & 10 & 9 \end{pmatrix}$$ Let's compute the matrix multiplication. ### Step 2: Transpose of matrix $$B$$ The transpose of matrix $$B$$, denoted as $$B^T$$, is obtained by switching its rows and columns. $$B^T = \begin{pmatrix} 7 & -1 & -2 \\ 13 & -7 & 2 \\ -6 & 2 & 0 \end{pmatrix}$$ ### Step 3: Compute $$A^2 - B^T$$ We now subtract $$B^T$$ from $$A^2$$. ### Step 4: Multiply $$(A^2 - B^T)$$ by vector $$C$$ Finally, we will multiply the resulting matrix by $$C$$. I'll calculate each of these steps to give the final result.The result of $$(A^2 - B^T)C$$ is: $$\begin{pmatrix} 162 \\ 90 \\ 420 \end{pmatrix}$$ Let me know if you'd like more details or if you have any questions! Here are 5 related questions to further explore: 1. How do you compute the determinant of matrix $$A$$? 2. What is the inverse of matrix $$A$$ (if it exists)? 3. How do you verify the correctness of a matrix multiplication step? 4. How would the result change if $$C$$ was a different vector? 5. What are the properties of matrix transposition? **Tip**: When multiplying matrices, always check the dimensions to ensure compatibility for matrix multiplication!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Matrix Transposition
Matrix Subtraction
Vector Multiplication
Formulas
Matrix multiplication: (AB)_{ij} = \sum_k A_{ik}B_{kj}
Matrix transposition: (B^T)_{ij} = B_{ji}
Matrix subtraction: A - B = A_{ij} - B_{ij}
Matrix-vector multiplication: (AC)_i = \sum_j A_{ij}C_j
Theorems
Properties of Matrix Transposition
Matrix Multiplication Rules
Suitable Grade Level
Undergraduate Math or Advanced High School (Grade 11-12)