Proving the Determinant Identity: [a b b c c a] 2 [a b c]

In this article, we explore the process of proving the determinant identity [a b b c c a] 2 [a b c]

Introduction to Determinants and Vectors

Determinants are a fundamental concept in linear algebra, used to analyze systems of linear equations and to understand various properties of matrices. This article will demonstrate how to prove a specific determinant identity involving vectors and matrices. We will use the properties of determinants and matrix operations to derive the desired result.

Step-by-Step Proof of the Identity

To prove that [a b b c c a] 2 [a b c], we will follow these steps:

Express the Determinant

The determinant of the vectors a, b, and c can be expressed as follows:

[a b c] det(A)

where A is a matrix formed by the vectors a, b, and c as its columns.

Form the New Matrix

We need to form the matrix B with columns a b, b c, and c a:

B [a b b c c a]

Rewrite the New Columns

We can express the columns of matrix B in terms of a combination of the original vectors:

B [1 1 0; 0 1 1; 1 0 1] [a b c]

This means we can rewrite the columns of B as:

B [1 0 1; 1 1 0; 0 1 1] [a b c]

Calculate the Determinant

We can use the linearity of the determinant to simplify this expression:

[a b b c c a] det([1 1 0; 0 1 1; 1 0 1]) * [a b c]

Use Properties of Determinants

The determinant of a product of matrices is the product of their determinants. Thus:

[a b b c c a] det([1 1 0; 0 1 1; 1 0 1]) * [a b c]

Calculate the Determinant of the Transformation Matrix

The determinant of the transformation matrix can be calculated as follows:

det([1 1 0; 0 1 1; 1 0 1]) 1*1*1 - 1*0 - 1*0 - 1*1 * 0*1*0 - 1 2

This means that:

[a b b c c a] 2[a b c]

Conclusion

This proves the required identity. Therefore, we have:

[a b b c c a] 2[a b c]

Thus, we see that the determinant [a b b c c a] is indeed twice the determinant [a b c].

Additional Insights

The process of proving this identity provides valuable insights into how determinants and vector transformations interact. It highlights the importance of matrix properties and their applications in solving complex problems in linear algebra.

Final Answer

Thus, the final answer is 2 [a b c]

Conclusion

In conclusion, we have demonstrated the process of proving the determinant identity using the properties of determinants and matrix operations. This proof not only verifies the identity but also reinforces our understanding of linear algebra concepts.