Question

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

• A. $-1$
• B. $2$
• C. $1$
• D. $-\sqrt 2$

Answer 1

Answer: (B)

$2$

Answer 2

$|(A+I)(adj\ A+I)|=4$
$⇒|A\ adj\ A +A+adj \ A+I|=4$
$⇒|(A)I+A+adj\ A+I|=4$
$|A|=−1$
$⇒|A+adj\ A|=4$
$A=\begin{bmatrix} a & b\\\[0.3em] c & d \\\[0.3em] \end{bmatrix}$

$adj A=\begin{bmatrix} a & -b\\\[0.3em] -c & d \\\[0.3em] \end{bmatrix}$

$⇒ \begin{bmatrix} (a+d) & b\\\[0.3em] 0 & (a+d) \\\[0.3em] \end{bmatrix}=4$
$⇒ a + d = ±2$

So, the correct option is (B): $2$