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Matrix and Determinant
Let $a,b,c,d,t$ is real number.
if $ A = \begin{bmatrix}
a & b \\
c&d
\end{bmatrix}$ and $det(A) = t \neq0$
and $det(A+t^{2}A^{-1})=0$
Find $det(A-t^{2}A^{-1})?$
Sum identity for 2×2 matrices
$\det(A+B)=\det(A)+\det(B)+\operatorname{tr}(A)\operatorname{tr}(B)-\operatorname{tr}(AB)$
$A=B,~\operatorname{tr}(A^2)=\operatorname{tr}(A)^2-2\det(A)$
$\det(A+t^2 A^{-1})=0$
$\det(A^2+t^2 I)=t^2+t^4+2t^2\operatorname{tr}(A^2)-t^2\operatorname{tr}(A^2)=0$
$(t-1)^2+\operatorname{tr}(A)^2=0$
As $\det(A),\operatorname{tr}(A)\in \mathbb{R}$
$t=1,~\operatorname{tr}(A)=0,~\operatorname{tr}(A^2)=-2$
$\det(A-t^2 A^{-1})=\det(A^2-t^2 I)$
$=t^2+t^4-2t^2\operatorname{tr}(A^2)+t^2\operatorname{tr}(A^2)=4$ |
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