Search the corpus

    Invertibility of $I - BA$ from that of $I - AB$

    E-HK-62-8

    Area
    Algebra
    Topics
    Invertibility, Linear Algebra
    Status
    draft

    Let \(A\) and \(B\) be \(n \times n\) matrices over the field \(F\). Prove that if \((I - AB)\) is invertible, then \(I - BA\) is invertible and \[ (I - BA)^{-1} = I + B(I - AB)^{-1}A. \]

    Solution

    Set \(C=I+B(I-AB)^{-1}A\). Since \((I-BA)B=B(I-AB)\), \[ (I-BA)C=I-BA+B(I-AB)(I-AB)^{-1}A=I. \] Also \(A(I-BA)=(I-AB)A\), so \[ C(I-BA)=I-BA+B(I-AB)^{-1}(I-AB)A=I. \] Thus \(C=(I-BA)^{-1}\).

    Source Collections