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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
Idempotent
Involutary
Non singular
Nilpotent
Solution
$a = + 2$ ; $b = – 4 $;$ c = 1$ ;$ d = – 2$
Let $A =$$\left[ {\begin{array}{*{20}{c}}2&{ – 4}\\1&{ – 2}\end{array}} \right]$
now $\left[ {\begin{array}{*{20}{c}}2&{ – 4}\\1&{ – 2}\end{array}} \right]$ $\left[ {\begin{array}{*{20}{c}}2&{ – 4}\\1&{ – 2}\end{array}} \right]$ $=$ $\left[ {\begin{array}{*{20}{c}}0&0\\ 0&0\end{array}} \right]$ $=$ null matrix
hence $A$ is nilpotent
note that any matrix of the form $\left[ {\begin{array}{*{20}{c}} a&{ – {a^2}}\\1&{ – a}\end{array}} \right]$ is a nilpotent
