If A = $\left[\begin{array}{ccc}6& 8& 5\\ 4& 2& 3\\ 9& 7& 1\end{array}\right]$ is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is :

• $\left[\begin{array}{ccc}6& 6& 7\\ 6& 2& 5\\ 7& 5& 1\end{array}\right]$

• $\left[\begin{array}{ccc}0& 2& - 2\\ - 2& 5& - 2\\ 2& 2& 0\end{array}\right]$

• $\left[\begin{array}{ccc}6& 6& 7\\ - 6& 2& - 5\\ - 7& 5& 1\end{array}\right]$

• $\left[\begin{array}{ccc}0& 6& - 2\\ 2& 0& - 2\\ - 2& - 2& 0\end{array}\right]$

If A = $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, then A¹⁰⁰ is equal to

• 2¹⁰⁰A

• 2⁹⁹A

• 100A

• 299A

If $\left[\begin{array}{ccc}1& 1& 1\\ 1& - 2& - 2\\ 1& 3& 1\end{array}\right]\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right] = \left[\begin{array}{c}0\\ 3\\ 4\end{array}\right], \mathrm{then} \left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$ is equal to :

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

• $\left[\begin{array}{c}1\\ 2\\ - 3\end{array}\right]$

• $\left[\begin{array}{c}5\\ - 2\\ 1\end{array}\right]$

• $\left[\begin{array}{c}1\\ - 2\\ 3\end{array}\right]$

If A and B are 2 x 2 matrices, then which of the following is true ?

• (A + B)² = A² + B² + 2AB

• (A - B)² = A² + B² - 2AB

• (A - B)(A + B) = A² + AB - BA - B²

• (A - B)(A + B) = A² - B²

If w is an imaginary root of unity, then the $\left|\begin{array}{ccc}\mathrm{a}& {\mathrm{bw}}^2& \mathrm{aw}\\ \mathrm{bw}& \mathrm{c}& {\mathrm{bw}}^2\\ {\mathrm{cw}}^2& \mathrm{aw}& \mathrm{c}\end{array}\right|$ is :

• a³ + b³ + c³

• a²b - b²c

• 0

• a³ + b³ + c³ - 3abc

If A = {x, y}, then the power set of A is :

• {x^y, y^x}

• $\left\{\mathrm{\varphi }, \mathrm{x}, \mathrm{y}\right\}$

• $\left\{\mathrm{\varphi }, \left\{\mathrm{x}\right\}, \left\{2\mathrm{y}\right\}\right\}$

• $\left\{\mathrm{\varphi }\left\{\mathrm{x}\right\}, \left\{\mathrm{y}\right\}, \left\{\mathrm{x}, \mathrm{y}\right\}\right\}$

397.

Let A, B and C be n x n matrices. Which one of the following is a correct statement?

• If AB = AC, then B = C

• If A³ + 2A² + 3A + 51 = 0, then A is invertible

• If A² = 0,then A = 0

• None of these

If A = $\left[\begin{array}{ccc}2& 4& 5\\ 4& 8& 10\\ - 6& - 12& - 15\end{array}\right]$, then rank of A is equal to :

• 0

• 1

• 2

• 3

What must be the matrix X if $2\mathrm{X} + \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right] = \left[\begin{array}{cc}3& 8\\ 7& 2\end{array}\right]$ ?

• $\left[\begin{array}{cc}1& 3\\ 2& - 1\end{array}\right]$

• $\left[\begin{array}{cc}1& - 3\\ 2& - 1\end{array}\right]$

• $\left[\begin{array}{cc}2& 6\\ 4& - 2\end{array}\right]$

• $\left[\begin{array}{cc}2& - 6\\ 4& - 2\end{array}\right]$

Inverse of the matrix $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& - \sin \left(2\mathrm{\theta }\right)\\ \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\end{array}\right]$ is

• $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& - \sin \left(2\mathrm{\theta }\right)\\ \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& \sin \left(2\mathrm{\theta }\right)\\ \sin \left(2\mathrm{\theta }\right)& - \cos \left(2\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& \sin \left(2\mathrm{\theta }\right)\\ \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& \sin \left(2\mathrm{\theta }\right)\\ - \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\end{array}\right]$