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21.
Let Q = $(\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) \\ \sin (\frac{π}{4}) & \cos (\frac{π}{4}) \end{matrix}) and x = (\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix})$ , then Q³x is equal to

$(\begin{matrix} 0 \\ 1 \end{matrix})$

$(\begin{matrix} - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix})$

$(\begin{matrix} - 1 \\ 0 \end{matrix})$

$(\begin{matrix} - \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} \end{matrix})$

$(\begin{matrix} - 1 \\ 0 \end{matrix})$

Given,

Q = $[\begin{matrix} \cos (\frac{π}{4}) & - \sin (\frac{π}{4}) \\ \sin (\frac{π}{4}) & \cos (\frac{π}{4}) \end{matrix}] and x = [\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}]$

Let $Q (θ) [\begin{matrix} \cos (θ) & - \sin (θ) \\ \sin (θ) & \cos (θ) \end{matrix}]$

$∵ Q^{3} (θ) = Q (3 θ)$

$∴ Q^{3} (\frac{π}{4}) = [\begin{matrix} \cos (\frac{3 π}{4}) & - \sin (\frac{3 π}{4}) \\ \sin (\frac{3 π}{4}) & \cos (\frac{3 π}{4}) \end{matrix}]$

= $[\begin{matrix} \frac{- 1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{matrix}]$

Now, $Q^{3} (\frac{π}{4}) . x = [\begin{matrix} \frac{- 1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & - \frac{1}{\sqrt{2}} \end{matrix}] [\begin{matrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix}] = [\begin{matrix} - \frac{1}{2} - \frac{1}{2} \\ \frac{1}{2} - \frac{1}{2} \end{matrix}]$

$∴ Q^{3} (\frac{π}{4}) x = [\begin{matrix} - 1 \\ 0 \end{matrix}]$

22.

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

a = 2n, b = 2ⁿ
a = 2ⁿ, b = 2n
a = 2ⁿ, b = n2^{n - 1}
a = 2ⁿ, b = n2ⁿ

23.

If $f (x) = |\begin{matrix} 1 & x & x + 1 \\ 2 x & x (x - 1) & (x + 1) x \\ 3 x (x - 1) & x (x - 1) (x - 2) & (x + 1) x (x - 1) \end{matrix}|$

Then, f(100) is equal to

24.

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

25.

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

26.

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