If A = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\alpha }\right)& \cos \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\alpha }\right)\\ \cos \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\alpha }\right)& {\sin }^2\left(\mathrm{\alpha }\right)\end{array}\right]$ and B = $\left[\begin{array}{cc}{\cos }^2\left(\mathrm{\beta }\right)& \cos \left(\mathrm{\beta }\right)\sin \left(\mathrm{\beta }\right)\\ \cos \left(\mathrm{\beta }\right)\sin \left(\mathrm{\beta }\right)& {\sin }^2\left(\mathrm{\beta }\right)\end{array}\right]$ are two matrices such that the product AB is null matrix, then $\mathrm{\alpha } - \mathrm{\beta }$ is

• 0

• multiple of $\mathrm{\pi }$

• an odd multiple of $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• None of these

If A is a square matrix of order n x n, then adj (adj A) is equal to

• ${\left|\mathrm{A}\right|}^{\mathrm{n}}\mathrm{A}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 1}\mathrm{A}$

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

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

For the equations x + 2y + 3z = 1, 2x + y + 3z = 2 and 5x + 5y + 9z = 4

• there is only one solution

• there exists infinitely many solution

• there is no solution

• None of the above

The inverse matrix of $\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$, is

• $\left[\begin{array}{ccc}\frac{1}{2}& - \frac{1}{2}& \frac{1}{2}\\ - 4& 3& - 1\\ \frac{5}{2}& - \frac{3}{2}& \frac{1}{2}\end{array}\right]$

• $\left[\begin{array}{ccc}\frac{1}{2}& - 4& \frac{5}{2}\\ 1& - 6& 3\\ 1& 2& - 1\end{array}\right]$

• $\frac{1}{2}\left[\begin{array}{ccc}1& 2& 3\\ 3& 2& 1\\ 4& 2& 3\end{array}\right]$

• $\frac{1}{2}\left[\begin{array}{ccc}1& - 1& - 1\\ - 8& 6& - 2\\ 5& - 3& 1\end{array}\right]$

355.

If A = $\left[\begin{array}{ccc}3& 2& 4\\ 1& 2& 1\\ 3& 2& 6\end{array}\right]$ and A_ij are the cofactors of a_ij then a₁₁A₁₁ + a₁₂A₁₂ + a₁₃A₁₃ is equal to

• 8

• 6

• 4

• 0

A = $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$ and AB = BA = I, then B is equal to

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

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

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

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

Let A = $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& - \cos \left(\mathrm{\theta }\right)\end{array}\right]$, then the inverse of A is

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

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

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

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

If matrix $\mathrm{A} = \left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right]$, then ${\left|\mathrm{A}\right|}^{- 1}$ is equal to

• ab - bc

• $\frac{1}{\mathrm{ab} - \mathrm{bc}}$

• $\frac{1}{\mathrm{ab} - \mathrm{bc}}\left[\begin{array}{cc}\mathrm{d}& - \mathrm{b}\\ - \mathrm{c}& \mathrm{a}\end{array}\right]$

• None of these

The multiplicative inverse of A = $\left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& - \sin \left(\mathrm{\theta }\right)\\ \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$ is

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

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

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

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

If A = $\left[\begin{array}{ccc}1& 1& 0\\ 2& 1& 5\\ 1& 2& 1\end{array}\right]$, then a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ is equal to

• 1

• 0

• - 1

• 2