Math Problem Statement
If A = [[1, 6], [-7, 5]], B = [[6], [-7]], C = [[8], [-11]], then find X when X = A^2 - 2BC.
Solution
The problem in the image involves matrix operations. The given matrices are:
You are asked to find where .
Step 1: Calculate
First, we need to calculate , which is .
Multiply the matrices:
(1 \times 1 + 6 \times -7) & (1 \times 6 + 6 \times 5) \\ (-7 \times 1 + 5 \times -7) & (-7 \times 6 + 5 \times 5) \end{bmatrix}$$ $$A^2 = \begin{bmatrix} 1 - 42 & 6 + 30 \\ -7 - 35 & -42 + 25 \end{bmatrix}$$ $$A^2 = \begin{bmatrix} -41 & 36 \\ -42 & -17 \end{bmatrix}$$ ### Step 2: Calculate $$2BC$$ Next, calculate $$2BC$$: $$2BC = 2 \times \left(\begin{bmatrix} 6 \\ -7 \end{bmatrix} \times \begin{bmatrix} 8 & -11 \end{bmatrix}\right)$$ The multiplication of $$B$$ and $$C$$ gives: $$BC = \begin{bmatrix} 6 \times 8 & 6 \times -11 \\ -7 \times 8 & -7 \times -11 \end{bmatrix} = \begin{bmatrix} 48 & -66 \\ -56 & 77 \end{bmatrix}$$ Now multiply by 2: $$2BC = \begin{bmatrix} 96 & -132 \\ -112 & 154 \end{bmatrix}$$ ### Step 3: Calculate $$X = A^2 - 2BC$$ $$X = \begin{bmatrix} -41 & 36 \\ -42 & -17 \end{bmatrix} - \begin{bmatrix} 96 & -132 \\ -112 & 154 \end{bmatrix}$$ Subtract the matrices: $$X = \begin{bmatrix} -41 - 96 & 36 + 132 \\ -42 + 112 & -17 - 154 \end{bmatrix}$$ $$X = \begin{bmatrix} -137 & 168 \\ 70 & -171 \end{bmatrix}$$ ### Final Answer $$X = \begin{bmatrix} -137 & 168 \\ 70 & -171 \end{bmatrix}$$ Would you like any further details or clarifications on this problem? Here are five related questions you might be interested in: 1. How do you perform matrix multiplication for larger matrices? 2. What is the significance of the determinant in matrix operations? 3. Can matrices with different dimensions be multiplied? 4. How do you find the inverse of a matrix? 5. What are some real-world applications of matrix operations? **Tip:** When performing matrix operations, always double-check the dimensions to ensure that the multiplication is valid.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Matrix Subtraction
Algebra
Formulas
Matrix Multiplication: (AB)_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}
Matrix Subtraction: (A - B)_{ij} = A_{ij} - B_{ij}
Theorems
Properties of Matrices
Suitable Grade Level
Undergraduate Level or Advanced High School