Let $A$ and $B$ be any two $3\times3$ matrices. If $A$ is symmetric and $B$ is skewsymmetric, then the matrix $AB – BA$ is

Let $A$ be a square matrix such that $A A^T=I$. Then $\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]$ is equal to

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 ${\rm{tr}}\left( A \right) \ne 0$

If $AA^T = I$ and $C$ is skew symmetric matric then $((A^T CA)^{50})^T$ is equal to

Let $a =$$\mathop {Lim}\limits_{x \to 1} \,\,\frac{x}{{\ln \,x}}\; – \;\frac{1}{{x\,\ln \,x}}$ ; $b =$$\mathop {Lim}\limits_{x \to 0} \,\,\frac{{{x^3} – 16x}}{{4x + {x^2}}}$ ; $c =$$\mathop {Lim}\limits_{x \to 0} \,\,\frac{{\ln (1 + \sin x)}}{x}$ and $d =$$\mathop {Lim}\limits_{x \to  – 1} \,\,\frac{{{{(x + 1)}^3}}}{{3\left( {\sin (x + 1) – (x + 1)} \right)}}$ , then the matrix $\left[ {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right]$ is