"Let A be a 2 $\times$ 2 matrix with det (A) = $-$ 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :"

Question

Accepted Answer

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$|(A + I)(adj\,A + I)| = 4$

$\Rightarrow |A\,adj\,A + A + adj\,A + I| = 4$

$\Rightarrow |(A)I + A + adj\,A + I| = 4$

$|A| = - 1 \Rightarrow |A + adj\,A| = 4$

$$A = \left[ {\matrix{ a & b \cr c & d \cr } } \right]\,adj\,A = \left[ {\matrix{ a & { - b} \cr { - c} & d \cr } } \right]$$

$$ \Rightarrow \left| {\matrix{ {(a + d)} & 0 \cr 0 & {(a + d)} \cr } } \right| = 4$$

$\Rightarrow a + d = \, \pm \,2$

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Let A be a 2 $\times$ 2 matrix with det (A) = $-$ 1 and det ((A + I) (Adj (A) + I)) = 4. Then the sum of the diagonal elements of A can be :