If $\bar{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}, \quad \bar{b}=b_1 \hat{i}+b_2 \hat{j}+b_3 \hat{k}$, $\bar{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ and $\left[\begin{array}{lll}3 \bar{a}+\bar{b} & 3 \bar{b}+\bar{c} & 3 \bar{c}+\bar{a}\end{array}\right]=\lambda\left|\begin{array}{lll}\overline{\mathrm{a}} \cdot \hat{\mathrm{i}} & \overline{\mathrm{a}} \cdot \hat{\mathrm{j}} & \overline{\mathrm{a}} \cdot \hat{\mathrm{k}} \\ \overline{\mathrm{b}} \cdot \hat{\mathrm{i}} & \overline{\mathrm{b}} \cdot \hat{\mathrm{j}} & \overline{\mathrm{b}} \cdot \hat{\mathrm{k}} \\ \overline{\mathrm{c}} \cdot \hat{\mathrm{i}} & \overline{\mathrm{c}} \cdot \hat{\mathrm{j}} & \overline{\mathrm{c}} \cdot \hat{\mathrm{k}}\end{array}\right|,$ then the value of $\lambda$ is

Let $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right], x \in \mathbb{R}^{+}$and $A^4=\left[a_{i j}\right]_2$. If $a_{11}=109$, then $\left(A^4\right)^{-1}=$

If $A=\left[\begin{array}{cc}5 a & -b \\ 3 & 2\end{array}\right]$ and $A \cdot \operatorname{adj} A=A A^T$, then $5 a+b$ is equal to

Let A and B be $3 \times 3$ real matrices such that A is symmetric matrix and B is skew-symmetric matrix. Then the system of linear equations $\left(A^2 B^2-B^2 A^2\right) X=O$. where $X$ is $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix, has