If A = $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& - \sin \left(2\mathrm{\theta }\right)\\ \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\end{array}\right]$ and A + A^T = I, where I is the unit matrix of 2 x 2 and A^T is the transpose of A, then the value of $\mathrm{\theta }$ is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\mathrm{\pi }$

• $\frac{3\mathrm{\pi }}{2}$

If A = $\left[\begin{array}{cc}3& 1\\ - 1& 2\end{array}\right]$, then A² - 5A is equal to

• I

• - I

• 7I

• - 7I

Let $∆ = \left|\begin{array}{ccc}\mathrm{Ax}& {\mathrm{x}}^2& 1\\ \mathrm{By}& {\mathrm{y}}^2& 1\\ \mathrm{Cz}& {\mathrm{z}}^2& 1\end{array}\right| \mathrm{and} {∆}_1 = \left|\begin{array}{ccc}\mathrm{A}& \mathrm{B}& \mathrm{C}\\ {\mathrm{x}}^2& {\mathrm{y}}^2& {\mathrm{z}}^2\\ 1& 1& 1\end{array}\right|, \mathrm{then} \left|\begin{array}{ccc}\mathrm{Ax}& \mathrm{By}& \mathrm{Cz}\\ {\mathrm{x}}^2& {\mathrm{y}}^2& {\mathrm{z}}^2\\ 1& 1& 1\end{array}\right|$

• ${∆}_1 = 2∆$

• ${∆}_1 = - ∆$

• ${∆}_1 = ∆$

• ${∆}_1 \ne ∆$

454.

If $\left|\begin{array}{cc}3& \mathrm{x}\\ \mathrm{x}& 1\end{array}\right| = \left|\begin{array}{cc}3& 2\\ 4& 1\end{array}\right|$, then x is equal to

• 4

• 8

• 2

• $\pm 2\sqrt{2}$

If $2\left|\begin{array}{cc}1& 3\\ 0& \mathrm{x}\end{array}\right| + \left|\begin{array}{cc}\mathrm{y}& 0\\ 1& 2\end{array}\right| = \left|\begin{array}{cc}5& 6\\ 1& 8\end{array}\right|$, then the value of x and y are

• x = 3, y = 3

• x = - 3, y = 3

• x = 3, y = - 3

• x = - 3, y = - 3

If A is a square matrix of order 3 x 3, then $\left|\mathrm{KA}\right|$ is equal to

• ${\mathrm{k}}^2\left|\mathrm{A}\right|$

• $\mathrm{K}\left|\mathrm{A}\right|$

• $3\mathrm{K}\left|\mathrm{A}\right|$

• ${\mathrm{k}}^3\left|\mathrm{A}\right|$

If A = $\frac{1}{\mathrm{\pi }}\left|\begin{array}{cc}{\sin }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)& {\cot }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right|, \mathrm{B} = \left|\begin{array}{cc}- {\cos }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)& - {\tan }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right|$, then A - B is

• 0

• $\frac{1}{2}\mathrm{I}$

• I

• 2I

If a matrix A is both symmetric and skew symmetric, then

• A is diagonal matrix

• A is a zero matrix

• A is scalar matrix

• A is square matrix

If $\left|\begin{array}{ccc}6\mathrm{i}& 3\mathrm{i}& - 1\\ 4& 3\mathrm{i}& - 1\\ 20& 3& \mathrm{i}\end{array}\right|$ = x + iy, then

• x = 3, y = 1

• x = 1, y = 3

• x = 0, y = 3

• x = 0, y = 0

$\left|\begin{array}{cc}1& {\log }_{\mathrm{b}}\left(\mathrm{a}\right)\\ {\log }_{\mathrm{a}}\left(\mathrm{b}\right)& 1\end{array}\right|$ is equal to

• 1

• 0

• ${\log }_{\mathrm{a}}\left(\mathrm{b}\right)$

• ${\log }_{\mathrm{b}}\left(\mathrm{a}\right)$