If A is a non-zero square matrix of order n with det (I + A) $\ne$ 0 and A³ = 0, where I, 0 are unit and null matrices of order n x n respectively, then (I + A)^{- 1} is equal to

• I - A + A²

• I + A + A²

• I + A ^{- 1}

• I + A

$\mathrm{If} \mathrm{x} \mathrm{is} \mathrm{real}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \frac{{\mathrm{x}}^2 - 3\mathrm{x} + 4}{{\mathrm{x}}^2 + 3\mathrm{x} + 4} \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

• $\left[\frac{1}{3}, 3\right]$

• $\left[\frac{1}{5}, 5\right]$

• $\left[\frac{1}{6}, 6\right]$

• $\left[\frac{1}{7}, 7\right]$

$\mathrm{A}\left(\mathrm{\alpha }, \mathrm{\beta }\right) = \left[\begin{array}{ccc}\cos \left(\mathrm{\alpha }\right)& \sin \left(\mathrm{\alpha }\right)& 0\\ - \sin \left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)& 0\\ 0& 0& {\mathrm{e}}^{\mathrm{\beta }}\end{array}\right] \Rightarrow {\left[\mathrm{A}\left(\mathrm{\alpha }, \mathrm{\beta }\right)\right]}^{ - 1} =$

• $\mathrm{A}\left( - \mathrm{\alpha }, \mathrm{\beta }\right)$

• $\mathrm{A}\left( - \mathrm{\alpha }, - \mathrm{\beta }\right)$

• $\mathrm{A}\left(\mathrm{\alpha }, - \mathrm{\beta }\right)$

• $\mathrm{A}\left(\mathrm{\alpha }, \mathrm{\beta }\right)$

$$\begin{gathered} \mathrm{If} \mathrm{A} \mathrm{is} \mathrm{a} \mathrm{matrix} \mathrm{such} \\ \left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]\mathrm{A}\left[\begin{array}{cc}1& 1\end{array}\right] = \left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right] \mathrm{then} \mathrm{A} = ? \end{gathered}$$

• $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}2& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 0\\ - 1& 1\end{array}\right]$

• $\left[\begin{array}{c}2\\ - 3\end{array}\right]$

$\mathrm{A} = \left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 1& 0\end{array}\right] \Rightarrow {\mathrm{A}}^2 - 2\mathrm{A} =?$

• A ^{- 1}

• - A ^{- 1}

• I

• - I

$\left|\begin{array}{ccc}24& 25& 26\\ 25& 26& 27\\ 26& 27& 27\end{array}\right| = ?$

• 0

• - 1

• 1

• 2

307.

$\mathrm{A} = \left[\begin{array}{cc}\mathrm{i}& - \mathrm{i}\\ - \mathrm{i}& \mathrm{i}\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}1& - 1\\ - 1& 1\end{array}\right] \Rightarrow {\mathrm{A}}^8$

• 4B

• 8B

• 64B

• 128B

$$\begin{gathered} \mathrm{Let} \mathrm{A} = \left[\begin{array}{ccc}- 1& - 2& - 3\\ 3& 4& 5\\ 4& 5& 6\end{array}\right], \mathrm{B} = \left[\begin{array}{cc}1& - 2\\ - 1& 2\end{array}\right] \mathrm{and} \mathrm{C} = \left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right], \mathrm{if} \mathrm{a}, \mathrm{b} \mathrm{and} \mathrm{c} \mathrm{respectively}, \\ \mathrm{denote} \mathrm{the} \mathrm{rank} \mathrm{of} \mathrm{A}, \mathrm{B}, \mathrm{and} \mathrm{C}, \mathrm{then} \mathrm{the} \mathrm{correct} \mathrm{order} \mathrm{of} \mathrm{these} \mathrm{number} \mathrm{is} \end{gathered}$$

• $\mathrm{a} < \mathrm{b} < \mathrm{c}$

• $\mathrm{c} < \mathrm{b} < \mathrm{a}$

• $\mathrm{b} < \mathrm{a} < \mathrm{c}$

• $\mathrm{a} < \mathrm{c} < \mathrm{b}$

$$\begin{gathered} \mathrm{If} \mathrm{I} \mathrm{is} \mathrm{the} \mathrm{identity} \mathrm{matrix} \mathrm{of} \mathrm{order} 2 \mathrm{and} \\ \mathrm{A} =\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right], \mathrm{then} \mathrm{for} \mathrm{n} \ge 1, \mathrm{mathematical} \mathrm{induction} \mathrm{gives} \end{gathered}$$

• ${\mathrm{A}}^{\mathrm{n}} = \mathrm{nA} - \left(\mathrm{n} - 1\right)\mathrm{I}$

• ${\mathrm{A}}^{\mathrm{n}} = \mathrm{nA} + \left(\mathrm{n} - 1\right)\mathrm{I}$

• ${\mathrm{A}}^{\mathrm{n}} = 2^{\mathrm{n}}\mathrm{A} - \left(\mathrm{n} +\hspace{0.17em}1\right)\mathrm{I}$

• ${\mathrm{A}}^{\mathrm{n}} = 2^{\mathrm{n} - 1}\mathrm{A} - \left(\mathrm{n} - 1\right)\mathrm{I}$

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}- 8& 5\\ 2& 4\end{array}\right] \mathrm{satisfies} \mathrm{the} \mathrm{equation} {\mathrm{x}}^2 + 4\mathrm{x} - \mathrm{p} = 0, \mathrm{then} \mathrm{p} \mathrm{is} \mathrm{equal} \mathrm{to}$

• 64

• 42

• 36

• 24