Let $\mathrm{S}=\left\{\mathrm{A}=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]: a, b, c, d \in\{0,1,2,3,4\}\right.$ and $\left.\mathrm{A}^2-4 \mathrm{~A}+3 \mathrm{I}=0\right\}$ be a set of $2 \times 2$ matrices. Then the number of matrices in S , for which the sum of the diagonal elements is equal to 4 , is :

Let $A=\left[\begin{array}{ccc}1 & 1 & 2 \\ -2 & 0 & 1 \\ 1 & 3 & 5\end{array}\right]$. Then the sum of all elements of the matrix $\operatorname{adj}\left(\operatorname{adj}\left(2(\operatorname{adj} \mathrm{~A})^{-1}\right)\right)$ is equal to:

The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$. If the sum of this A.P. is the cube of its last term, then its common difference is:

The number of ways, of forming a queue of 4 boys and 3 girls such that all the girls are not together, is: