Determinant of a Matrix Satisfying ( A^2 A - I )

In this article, we will delve into a specific problem pertaining to the determinant of a matrix ( A ) which satisfies the equation ( A^2 A - I ). We will explore the relationship between the matrix's square and its determinant, eigenvalues, and the possible values of the determinant.

Deriving the Determinant

Suppose ( A ) is a square matrix of degree ( n ) such that ( A^2 A - I ). Let's start by examining the implications of this equation.

Equation Simplification

From the given equation ( A^2 A - I ), we can derive additional information about the matrix ( A ). Multiplying both sides by ( A ) to the power of 2, we obtain:

( A^3 -I )

This implies that:

( det(A^3) det(-I) )

Since ( det(A^3) (det(A))^3 ) and ( det(-I) (-1)^n ), we have:

( (det(A))^3 (-1)^n )

Therefore, we have two cases to consider:

Case i: ( n ) is Odd

When ( n ) is odd, we get:

( (det(A))^3 -1 )

The solutions to this equation are:

( det(A) in {-1, e^{ipi/3}, e^{i5pi/3}} )

Case ii: ( n ) is Even

When ( n ) is even, we obtain:

( (det(A))^3 1 )

The solutions to this equation are:

( det(A) in {1, e^{i2pi/3}, e^{i4pi/3}} )

Understanding Eigenvalues

If ( lambda ) is an eigenvalue of ( A ), the eigenvalues satisfy the equation ( lambda^2 - lambda 1 0 ). The roots of this polynomial equation are the non-real cube roots of unity, which are conjugate to each other and their product is ( 1 ).

For a real matrix, the complex eigenvalues must come in conjugate pairs. Therefore, the determinant of ( A ) can only be ( 1 ) if ( n ) is even.

If ( A ) is a complex matrix, the determinant can be expressed in the form ( alpha^k beta^{n-k} ), where ( alpha ) and ( beta ) are the non-real cube roots of ( -1 ).

Generalizing to Higher Dimensions

The possible values of ( det(A) ) are the sixth roots of unity. Given the equation ( A^2 A - I ), we can derive that ( A^6 I ), implying:

( det(A^6) 1 )

The sixth roots of unity are the roots of the polynomial ( x^6 - 1 0 ), which can be factored as ( (x - 1)(x 1)(x^2 - x 1)(x^2 x 1) ).

For a ( 1 times 1 ) matrix, ( A [alpha] ), where ( alpha ) is a sixth root of unity. For a larger matrix, we can extend this construction using the direct sum ( A oplus I_{n-1} ). Similarly, the matrix ( A alpha I_n ) or ( A alpha^{-1} I_n ) can be used for even ( n ).

The determinant of the ( 2 times 2 ) case can be represented as the product of the eigenvalues, which are either ( alpha ) or ( alpha^{-1} ). The possible values of ( A ) are ( 1, alpha^2, alpha^{-2} ).

For a ( 3 times 3 ) matrix, the determinant can include the sixth root of unity ( -1 ) by using ( A alpha I_3 ), making it possible for all six sixth roots of unity to be achievable.

Conclusion

In conclusion, the matrix ( A ) satisfying ( A^2 A - I ) can have a determinant that is one of the sixth roots of unity. The specific values depend on the parity of ( n ) and the nature of the eigenvalues.