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Multiple Choice Questions

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

$(\begin{matrix} 0 \\ 1 \end{matrix})$
$(\begin{matrix} - \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{matrix})$
$(\begin{matrix} - 1 \\ 0 \end{matrix})$
$(\begin{matrix} - \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} \end{matrix})$

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}

6

0

1

2/3

Let

$U_{i} = (\begin{matrix} a_{i} \\ b_{i} \\ c_{i} \end{matrix}), where i = 1, 2, 3 ∴ {AU}_{1} = (\begin{matrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{matrix}) (\begin{matrix} a_{1} \\ b_{1} \\ c_{1} \end{matrix}) \Rightarrow (\begin{matrix} a_{1} \\ 2 a_{1} + b_{1} \\ 3 a_{1} + 2 b_{1} + c_{1} \end{matrix}) = (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}) [∵ {AU}_{1} = (\begin{matrix} 1 \\ 0 \\ 0 \end{matrix})]$

$∴ a_{1} = 1, 2 a_{1} + b_{1} = 0 \Rightarrow 2 + b_{1} = 0 \Rightarrow b_{1} = - 2 and 3 a_{1} + 2 b_{1} + c_{1} = 0 \Rightarrow 3 + 2 (- 2) + c_{1} = 0 \Rightarrow 3 - 4 + c_{1} = 0 \Rightarrow c_{1} = 1$

Similarly, AU₂ = $(\begin{matrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{matrix}) (\begin{matrix} a_{2} \\ b_{2} \\ c_{2} \end{matrix})$

$\Rightarrow (\begin{matrix} a_{2} \\ 2 a_{2} + b_{2} \\ 3 a_{2} + 2 b_{2} + c_{2} \end{matrix}) = (\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}) [∵ {AU}_{2} = (\begin{matrix} 2 \\ 3 \\ 0 \end{matrix})]$

$∴ a_{2} = 2 and 2 a_{2} + b_{2} = 3 \Rightarrow 2 \times 2 + b_{2} = 3 \Rightarrow 4 + b_{2} = 3 \Rightarrow b_{2} = - 1 and 3 a_{2} + 2 b_{2} + c_{2} = 0 \Rightarrow 3 \times 2 + 2 (- 1) + c_{2} = 0 \Rightarrow 6 - 2 + c_{2} = 0 \Rightarrow c_{2} = - 4$

Now, AU₃ = $(\begin{matrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 3 & 2 & 1 \end{matrix}) (\begin{matrix} a_{3} \\ b_{3} \\ c_{3} \end{matrix})$

$\Rightarrow (\begin{matrix} a_{3} \\ 2 a_{3} + b_{3} \\ 3 a_{3} + 2 b_{3} + c_{3} \end{matrix}) = (\begin{matrix} 2 \\ 3 \\ 0 \end{matrix}) [∵ {AU}_{3} = (\begin{matrix} 2 \\ 3 \\ 0 \end{matrix})]$

$∴ a_{3} = 2 and 2 a_{3} + b_{3} = 3 \Rightarrow 2 \times 2 + b_{3} = 3 \Rightarrow 4 + b_{3} = 3 \Rightarrow b_{3} = - 1 and 3 a_{3} + 2 b_{3} + c_{3} = 1 \Rightarrow 3 \times 2 + 2 (- 1) + c_{3} = 1 \Rightarrow 6 - 2 + c_{3} = 1 \Rightarrow c_{2} = - 3$

$∴ U = (\begin{matrix} 1 & 2 & 2 \\ - 2 & - 1 & - 1 \\ 1 & - 4 & - 3 \end{matrix}) \Rightarrow |U| = 1 (3 - 4) - 2 (6 + 1) + 2 (8 + 1) = - 1 - 14 + 18 = 3 Also, adj (U) = {[\begin{matrix} - 1 & - 7 & 9 \\ - 2 & - 5 & 6 \\ 0 & - 3 & 3 \end{matrix}]}^{T} = [\begin{matrix} - 1 & - 2 & 0 \\ - 7 & - 5 & - 3 \\ 9 & 6 & 3 \end{matrix}] ∴ U^{- 1} = \frac{1}{|U|} adj (U)$

$\Rightarrow U^{- 1} = [\begin{matrix} - \frac{1}{3} & - \frac{2}{3} & 0 \\ - \frac{7}{3} & - \frac{5}{3} & - 1 \\ 3 & 2 & 1 \end{matrix}]$

Hence, the sum of all elements of u^- ¹ is 0

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

P
I - P
2I + P
2I - P

27.

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

P
2P
- P
- 2P

28.

The system of linear equations $λ$ x + y + z = 3, x - y - 2z = 6, - x + y + z = µ has

infinite number of solutions for $λ \neq$ -1 and all µ
infinite number of solutions for $λ$ = - 1 and $μ$ = 3
no solution for $λ \neq - 1$
unique solution for $λ$ = - 1 and $μ$ = 3

29.

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

30.

If A = $[\begin{matrix} 3 & x - 1 \\ 2 x + 3 & x + 2 \end{matrix}]$ is a symmetric matrix, then the value of x is

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Multiple Choice Questions

24. For a matrix A = 100210321, if U1, U2, and U3. are 3 × 1 column matrices satisfying AU1 = 100, AU2 = 230, AU3 = 231 and U is 3 × 3 matrix whose columns are U1, U2, and U3. Then, sum of the elements of U- 1 6 0 1 2/3