Matrices

Let Q = $\left(\begin{array}{cc}\cos \left(\frac{\mathrm{\pi }}{4}\right)& - \sin \left(\frac{\mathrm{\pi }}{4}\right)\\ \sin \left(\frac{\mathrm{\pi }}{4}\right)& \cos \left(\frac{\mathrm{\pi }}{4}\right)\end{array}\right) \mathrm{and} \mathrm{x} = \left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$, then Q³x is equal to

• $\left(\begin{array}{c}0\\ 1\end{array}\right)$

• $\left(\begin{array}{c}- \frac{1}{\sqrt{2}}\\ \frac{1}{\sqrt{2}}\end{array}\right)$

• $\left(\begin{array}{c}- 1\\ 0\end{array}\right)$

• $\left(\begin{array}{c}- \frac{1}{\sqrt{2}}\\ - \frac{1}{\sqrt{2}}\end{array}\right)$

If the matricx A = $\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 2& 0& 2\end{array}\right]$, then Aⁿ = $\left[\begin{array}{ccc}\mathrm{a}& 0& 0\\ 0& \mathrm{a}& 0\\ \mathrm{b}& 0& \mathrm{a}\end{array}\right], \mathrm{n}\in \mathrm{N}$ where

• a = 2n, b = 2ⁿ

• a = 2ⁿ, b = 2n

• a = 2ⁿ, b = n2^{n - 1}

• a = 2ⁿ, b = n2ⁿ

If $\mathrm{f}\left(\mathrm{x}\right) = \left|\begin{array}{ccc}1& \mathrm{x}& \mathrm{x} + 1\\ 2\mathrm{x}& \mathrm{x}(\mathrm{x} - 1)& \left(\mathrm{x} + 1\right)\mathrm{x}\\ 3\mathrm{x}\left(\mathrm{x} - 1\right)& \mathrm{x}(\mathrm{x} - 1)\left(\mathrm{x} - 2\right)& \left(\mathrm{x} + 1\right)\mathrm{x}\left(\mathrm{x} - 1\right)\end{array}\right|$

Then, f(100) is equal to

• 0

• 1

• 100

• 10

For a matrix $\mathrm{A} = \left(\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right)$, if U₁, U₂, and U₃. are $3 \times 1$ column matrices satisfying ${\mathrm{AU}}_1 = \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right), {\mathrm{AU}}_2 = \left(\begin{array}{c}2\\ 3\\ 0\end{array}\right), {\mathrm{AU}}_3 = \left(\begin{array}{c}2\\ 3\\ 1\end{array}\right)$ and U is $3 \times 3$  matrix whose columns are U₁, U₂, and U₃.  Then, sum of the elements of U^{- 1}

• 6

• 0

• 1

• 2/3

Let I denote the 3 x 3 identity matrix and P be a matrix obtained by rearranging the columns of I. Then,

• there are six distinct choices for P and det (P) = 1

• there are six distinct choices for P and det (P) =  $\pm 1$

• there are more than one choices for P and some of them are not invertible

• there are more than one choices for P and P^{- 1} = I in each choice

If I = $\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right) \mathrm{and} \mathrm{P} = \left(\begin{array}{ccc}1& 0& 0\\ 0& - 1& 0\\ 0& 0& - 2\end{array}\right)$. Then, the matrix P³ + 2P² is equal to

• P

• I - P

• 2I + P

• 2I - P

If P = $\left(\begin{array}{ccc}2& - 2& - 4\\ - 1& 3& 4\\ 1& - 2& - 3\end{array}\right)$, then P⁵ is equal to

• P

• 2P

• - P

• - 2P

The system of linear equations $\mathrm{\lambda }$x + y + z = 3,  x - y - 2z = 6, - x + y + z = µ has

• infinite number of solutions for $\mathrm{\lambda } \ne$-1 and all µ

• infinite number of solutions for $\mathrm{\lambda }$ = - 1 and $\mathrm{\mu }$ = 3

• no solution for $\mathrm{\lambda } \ne - 1$

• unique solution for $\mathrm{\lambda }$ = - 1 and $\mathrm{\mu }$ = 3

If A and B are two matrices such that A + B and AB are both defined, then

• A and B can be any matrices

• A, B are square matrices not necessarily of the same order

• A, B are square matrices of the same order

• number of columns of A = number of rows of B

If A = $\left[\begin{array}{cc}3& \mathrm{x} - 1\\ 2\mathrm{x} +\hspace{0.17em}3& \mathrm{x} + 2\end{array}\right]$ is a symmetric matrix, then the value of x is

• 4

• 3

• - 4

• - 3