What if matrix AB = BA = -I.

Answered question

2022-03-03

What if matrix AB = BA = -I. Assume that matrices A,B and I are nxn size matrices.

Answer & Explanation

RizerMix

RizerMix

Expert2023-04-23Added 656 answers

Given that AB = BA = -I, where A, B, and I are n×n size matrices.
To solve this problem, we can start by taking the determinant of both sides of the equation AB = -I. Using the property that det(AB)=det(A)det(B), we have:

det(A)det(B)=det(AB)=det(-I)=(-1)n

where n is the size of the matrices.

Since det(-I)=(-1)n, we know that n must be even in order for det(-I) to be positive. Therefore, we can assume that n is even and that det(A) and det(B) are nonzero.

Next, we can take the determinant of both sides of the equation BA = -I:

det(B)det(A)=det(BA)=det(-I)=(-1)n

Since det(A) and det(B) are both nonzero, we can divide both sides of the equation AB=BA=-I by det(A)det(B) to obtain:

AB=BA=-1det(A)2I

Therefore, the matrices A and B commute with each other and are both invertible. Moreover, the inverse of each matrix is given by:

A-1=-1det(A)2B
B-1=-1det(B)2A

To see why these formulas hold, we can compute:

ABA-1=-1det(A)2IA-1=-1det(A)2A-1
BAA-1=-1det(A)2BA-1=-1det(A)2A-1

Since A and B commute with each other, we can use the same formulas to compute A-1 and B-1 in terms of each other. Specifically, we have:

A-1=-1det(B)2B
B-1=-1det(A)2A

Therefore, we have found explicit formulas for the inverses of A and B in terms of each other, and we have shown that the matrices A and B commute with each other.

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