If $\left|\begin{array}{ccc}\mathrm{x}& 3& 6\\ 3& 6& \mathrm{x}\\ 6& \mathrm{x}& 3\end{array}\right| = \left|\begin{array}{ccc}2& \mathrm{x}& 7\\ \mathrm{x}& 7& 2\\ 7& 2& \mathrm{x}\end{array}\right| = \left|\begin{array}{ccc}4& 5& \mathrm{x}\\ 5& \mathrm{x}& 4\\ \mathrm{x}& 4& 5\end{array}\right| = 0$, then x is equal to

• 9

• - 9

• 0

• - 1

If w be the complex cube root of unity and matrix H = $\left[\begin{array}{cc}\mathrm{w}& 0\\ 0& \mathrm{w}\end{array}\right]$, then H⁷⁰ is equal to

• 0

• - H

• H

• H²

Let A = $\left[\begin{array}{cc}{\mathrm{\alpha }}^2& 5\\ 5& - \mathrm{\alpha }\end{array}\right]$ and $\left|{\mathrm{A}}^{10}\right| = 1024$, then $\mathrm{\alpha }$ is equal to

• 2

• - 2

• 3

• - 3

If w $\ne$ 1 is a cube root of unity, then the value of

$\left|\begin{array}{ccc}1 + 2{\mathrm{w}}^{100} + {\mathrm{w}}^{200}& {\mathrm{w}}^2& 1\\ 1& 1 + {\mathrm{w}}^{100} + 2{\mathrm{w}}^{200}& \mathrm{w}\\ \mathrm{w}& {\mathrm{w}}^2& 2+ {\mathrm{w}}^{100} + {\mathrm{w}}^{200}\end{array}\right|$ is equal to

• 0

• 1

• w

• w²

If $\mathrm{\alpha }$ is a cube root of unity, then the value of

$\left|\begin{array}{ccc}\mathrm{\alpha }& {\mathrm{\alpha }}^3& {\mathrm{\alpha }}^5\\ {\mathrm{\alpha }}^3& {\mathrm{\alpha }}^5& \mathrm{\alpha }\\ {\mathrm{\alpha }}^3& \mathrm{\alpha }& {\mathrm{\alpha }}^3\end{array}\right|$ is equal to

• $3{\mathrm{\alpha }}^3$

• $3\left({\mathrm{\alpha }}^3 + {\mathrm{\alpha }}^6 + {\mathrm{\alpha }}^9\right)$

• $3\left(\mathrm{\alpha } + {\mathrm{\alpha }}^2 + {\mathrm{\alpha }}^3\right)$

• 0

If $\mathrm{f}\left(\mathrm{\alpha }\right) = \left|\begin{array}{ccc}1& \mathrm{\alpha }& {\mathrm{\alpha }}^2\\ \mathrm{\alpha }& {\mathrm{\alpha }}^2& 1\\ {\mathrm{\alpha }}^2& 1& \mathrm{\alpha }\end{array}\right|$, then $\mathrm{f}\left(\sqrt[3]{3}\right)$ is equal to

• 1

• - 4

• 4

• 2

If A = $\left[\begin{array}{ccc}3& 3& 3\\ 3& 3& 3\\ 3& 3& 3\end{array}\right]$, then A⁴ is equal to

• 27 A

• 81 A

• 243 A

• 729 A

68.

If B is a non-singular matrix and A is a square matrix such that B^-1AB exists, then det (B^-1AB) is equal to

• det(A^-1)

• det (B^-1)

• det(B)

• det(A)

If 1, w, w² are the cube roots of unity and if

, a² + b² is equal to

• 1 + w²

• w² - 1

• 1 + w

• (1 + w)²

If A = $\left|\begin{array}{ccc}1& 0& 0\\ \mathrm{x}& 1& 0\\ \mathrm{x}& \mathrm{x}& 1\end{array}\right| \mathrm{and} \mathrm{I} = \left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|$, then A³ - 4A² + 3A + I is equal to

• 3I

• I

• - I

• - 2I