Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\left[\begin{array}{lll}1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1\end{array}\right]$ where each of $a, b$, and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

Express the matrix $\mathrm{B}=\left[\begin{array}{rrr}2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3\end{array}\right]$ as the sum of a symmetric and a skew symmetric matrix.

If $\mathrm{A}^{\prime}=\left[\begin{array}{cc}3 & 4 \\ -1 & 2 \\ 0 & 1\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}-1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right],$ then verify that  $(\mathrm{A}+\mathrm{B})^{\prime}=\mathrm{A}^{\prime}+\mathrm{B}^{\prime}$

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A),$ the sum of diagonal entries of $a$. Assume that ${A^2} = I$ .

Statement $-1 :$ If $A \ne I,A \ne – I$ then $\det \left( A \right) = – 1$

Statement $-2 :$ If $A \ne I,A \ne – I$ then ${\rm{tr}}\left( A \right) \ne 0$

Characteristic equation of $A = \left| {\begin{array}{*{20}{c}}
2&3&0\\
1&2&5\\
3&{ – 1}&2
\end{array}} \right|$ is