Question

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

Answer 1

Given $A A^{T} = 4 I \rightarrow \left|\right. A \left|\right. = \pm 2$
$ \rightarrow A^{T} = \frac{4 a d j A}{\left|\right. A \left|\right.} \rightarrow a_{\text{ij}} = \frac{4}{\left|\right. A \left|\right.} c_{\text{ij}}$
$\Rightarrow \left|\right. A \left|\right. = - 2 \left(\right. \text{as } a_{\text{ij}} + 2 C_{\text{ij}} = 0 \left.\right)$
$\text{Now } \left|\right. A + 4 I \left|\right. = \left|\right. A + A A^{T} \left|\right. = - 2 \left|\right. I + A \left|\right.$
$\Rightarrow \left|\right. A + 4 I \left|\right.+2\left|\right. A + I \left|\right.=0$
$\Rightarrow 5 \lambda = 2$