Question

Let $A={\left[a_{ij}\right]}_{3\times 3}$ be a matrix such that $A{A}^T=4I$ and $a_{ij}+2c_{ij}=0$ (where $C_{ij}$ is the cofactor of $a_{ij}$ and $I$ is the unit matrix of order 3). If the determinents are related by $\left|\begin{array}{ccc}{a}_{11}+4& a_{12}& a_{13}\\ a_{21}& a_{22}+4& a_{23}\\ a_{31}& a_{32}& a_{33}+4\end{array}\right|+5\lambda \left|\begin{array}{ccc}{a}_{11}+1& a_{12}& a_{13}\\ a_{21}& a_{22}+1& a_{23}\\ a_{31}& a_{32}& a_{33}+1\end{array}\right|=0$ then $10\lambda =$

Answer 1

Answer: 4

Given $A{A}^T=4I\to |A|=\pm 2$
$\to A^T=\frac{4adjA}{|A|}\to a_{\text{ij}}=\frac{4}{|A|}{c}_{\text{ij}}$
$\Rightarrow |A|=-2(\text{as }{a}_{\text{ij}}+2C_{\text{ij}}=0)$
$\text{Now }|A+4I|=|A+A{A}^T|=-2|I+A|$
$\Rightarrow |A+4I|+2|A+I|=0$
$\Rightarrow 5\lambda =2$