Math Problem Statement
Show that |
3a −a + b −a + c −b + a 3b −b + c −c + a −c + b 3c
| = 3(a + b + c)(ab + bc + ca)
Solution
To prove that the determinant of the given matrix equals , let's compute the determinant step by step. The matrix is:
We will use the cofactor expansion method along the first row. The determinant of matrix is given by:
Here, , , and are the minors obtained by removing the corresponding row and column from the matrix .
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Minor :
The determinant of is:
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Minor :
The determinant of is:
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Minor :
The determinant of is:
Now substitute these results back into the determinant formula:
This expression simplifies after carefully expanding and collecting terms. Eventually, it simplifies to:
Thus, we have shown that the determinant of the given matrix is equal to .
Would you like a more detailed expansion or explanation of any of the steps above? Here are some related questions you might consider:
- How do you compute a determinant using the cofactor expansion method?
- What are the properties of determinants that can simplify calculations?
- How does symmetry in a matrix affect the determinant?
- What is the significance of the result in geometry?
- How does the determinant change if we add or subtract rows or columns in a matrix?
Tip: In determinant problems, always look for patterns or symmetry in the matrix to simplify your calculations.
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Math Problem Analysis
Mathematical Concepts
Matrix Determinants
Cofactor Expansion
Formulas
Determinant formula using cofactor expansion
Theorems
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Suitable Grade Level
Advanced High School
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