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    $\det(I + A) = 1 + \det A$ exactly for trace zero

    E-HK-53-6

    Area
    Algebra
    Topics
    Determinants, Linear Algebra
    Status
    draft

    Let \(A\) be a \(2 \times 2\) matrix over a field. Prove that \(\det(I + A) = 1 + \det A\) if and only if \(\operatorname{trace}(A) = 0\).

    Solution

    Write \[ A=\begin{bmatrix}a&b\\c&d\end{bmatrix}. \] Then \[ \det(I+A)=(1+a)(1+d)-bc =1+(a+d)+(ad-bc) =1+\operatorname{trace}(A)+\det A. \] Therefore \[ \det(I+A)=1+\det A \] holds exactly when \(\operatorname{trace}(A)=0\).

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