Math Problem Statement
explain why det F became 0
Solution
To solve the problem and find the value of the scalar given as , and to explain why , we need to calculate the determinants of matrices and .
Calculation of
Matrix is a diagonal matrix, which simplifies the determinant calculation. The determinant of a diagonal matrix is simply the product of the diagonal elements. Therefore,
Calculation of
Matrix is more complex. To find the determinant, we can perform row operations to simplify the matrix without changing the determinant, or calculate directly if simplification is tricky. Looking at matrix , we can check for any row operations that simplify the calculation:
1 & -2 & 4 & -3 & 1 \\ 0 & -1 & 2 & 4 & 2 \\ 1 & 6 & -12 & 9 & -3 \\ -3 & 6 & -12 & 9 & -3 \\ 0 & 0 & 12 & 1 & 1 \end{bmatrix}$$ We can immediately notice that the third row and the fourth row are proportional: $$\text{Row 4} = -3 \times (\text{Row 3}) + \text{Row 1}.$$ This relationship indicates that the rows are linearly dependent. A matrix with linearly dependent rows has a determinant of zero. Thus, $$\text{det}(F) = 0$$. ### Conclusion Given $$\text{det}(F) = 0$$ and $$\text{det}(D) = 576$$, the value of scalar $$a$$ is: $$a = \text{det}(F) - \text{det}(D) = 0 - 576 = -576.$$ If you have any questions or need further details, feel free to ask. Here are some related questions you might consider: 1. How would you calculate the determinant of a 5x5 matrix without any obvious simplifications? 2. What properties of determinants can be useful in simplifying the calculation of a determinant? 3. Can you identify situations where the determinant of a matrix might be zero without doing calculations? 4. How would the result change if the matrices $$F$$ and $$D$$ were modified slightly? 5. What is the geometric interpretation of the determinant being zero? **Tip:** When you encounter a complex determinant calculation, consider using row and column operations to simplify the matrix while preserving the determinant’s value. This often includes making zeros to form an upper or lower triangular matrix, making the determinant calculation straightforward as the product of diagonal entries.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Determinants
Linear Algebra
Formulas
det(F) = det(Matrix F)
det(D) = det(Matrix D)
Determinant of a diagonal matrix = product of diagonal elements
Theorems
Determinant of a diagonal matrix
Property of determinants: det of matrix with linearly dependent rows = 0
Suitable Grade Level
Undergraduate Linear Algebra
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