Question

Let $A=\left[a_{i j}\right]_{3 \times 3}$ be a square matrix such that $AA^{T}=4I,\left|A\right| < 0$ . If $\begin{vmatrix} a_{11}+4 & a_{12} & a_{13} \\ a_{21} & a_{22}+4 & a_{23} \\ a_{31} & a_{32} & a_{33}+4 \end{vmatrix}=5\lambda \left|A + I\right|$ . Then, $\lambda $ is equal to

• A. $\frac{4}{5}$
• B. $-\frac{4}{5}$
• C. $\frac{8}{5}$
• D. $-\frac{8}{5}$

Answer 1

Answer: (D)

$AA^{T}=4I\Rightarrow \left|A A^{T}\right|=\left|4 I\right|\Rightarrow \left|A\right|^{2}=64$
$\Rightarrow \left|A\right|=-8$ or $8$
As, $\left|A\right| < 0\Rightarrow \left|A\right|=-8$
$\begin{vmatrix} a_{11}+4 & a_{12} & a_{13} \\ a_{21} & a_{22}+4 & a_{23} \\ a_{31} & a_{32} & a_{33}+4 \end{vmatrix}=\left|A + 4 I\right|=\left|A + A A^{T}\right|$
$=\left|A\right|\left|I + A^{T}\right|$
$\Rightarrow -8\left|A + I\right|=5\lambda \left|A + I\right|$
$\Rightarrow \lambda =-\frac{8}{5}$