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Question Number 55374 by gunawan last updated on 23/Feb/19

$$\mathrm{Given}\mathrm{matrices}A\left(x\right)=\left[\begin{array}{cc}x-1& 3\\ 4& x+3\end{array}\right]\in M_2\left(\mathbb{R}\right).$$$$\mathrm{The}\mathrm{smallest}\det \left(A\left(x\right)\right)\mathrm{is}..$$

Answered by Joel578 last updated on 23/Feb/19

$$\mid A\left(x\right)\mid =(x-1)(x+3)-12$$$$=x^2+2x-15$$$$={(x+1)}^2-16$$$$\mathrm{So}\mathrm{the}\mathrm{smallest}\mid A\left(x\right)\mid \mathrm{is}-16,\mathrm{when}x=-1$$

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