Proving Matrix Determinant Identities: An In-depth Analysis

This article focuses on proving an identity concerning the determinant of an n x n matrix, specifically when the inverse of the matrix is given as a multiple of its transpose. This topic is crucial for understanding advanced linear algebra and matrix theory. We will break down the proof step by step, ensuring each step is clear and comprehensible.

Introduction to Matrix Determinant Identities

In linear algebra, the determinant of a matrix is a scalar value that can be computed from the elements of a square matrix and carries significant information about the matrix, including whether it is invertible. For an n x n matrix A, the determinant is denoted as det A.

Key identities that we will use in our proof include: det(AT) det(A) If det(A) ≠ 0, then det(A-1) 1 / det(A) det(kA) kn det(A) for any scalar k det(AB) det(A) det(B)

Proving the Identity det(A2) 1 / 2n

Given that a matrix A is invertible and its inverse is related to its transpose by the equation A-1 2AT, we aim to prove that det(A2) 1 / 2n.

Step-by-Step Proof

1. Start with the given information: A is an invertible matrix such that A-1 2AT. 2. Use the property of determinants for the inverse:

det(A-1) 1 / det(A)

Substituting the given relationship:

det(2AT) 1 / det(A)

Using the property det(kA) kn det(A) for any scalar k and noting that 2AT involves multiplying each row by 2:

2n det(AT) 1 / det(A)

Since det(AT) det(A):

2n det(A) 1 / det(A)

Rewriting the equation:

2n (det A)2 1

Therefore:

(det A)2 1 / 2n

And:

det(A2) 1 / 2n

Alternative Approach

Another way to approach the problem is by leveraging the identity AAT 1/2 In, where In is the identity matrix of size n x n:

1. Given:

In A A-1 2A AT

Dividing both sides by 2:

1/2 In A AT

2. Now compute the determinant on both sides:

det(1/2 In) det(A AT)

Using det(kA) kn det(A) on the left side:

1/2n det(In) det(A) det(AT)

Since det(In) 1 and det(A) det(AT):

1/2n (det A)2

Therefore:

(det A)2 1 / 2n

And:

det(A2) 1 / 2n

Conclusion

Through detailed steps and leveraging key determinant identities, we have successfully proven that for an invertible matrix A where A-1 2AT, the determinant of A2 is 1/2n. This identity is fundamental in advanced matrix theory and possesses wide-ranging implications in mathematics and its applications.

Key Takeaways

The determinant of a matrix is a crucial scalar value that reflects the matrix's properties. Using determinant properties, such as det(A-1) 1 / det(A) and det(kA) kn det(A), can simplify complex problems. The transpose of a matrix does not change its determinant, i.e., det(AT) det(A). The identity matrix In has a determinant of 1.

Related Keywords

Matrix determinant Invertible matrix Transpose matrix