Download App

3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
Let $\alpha$ be a root of the equation $x^{2}+x+1=0$ and the matrix $A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}{1} & {1} & {1} \\ {1} & {\alpha} & {\alpha^{2}} \\ {1} & {\alpha^{2}} & {\alpha^{4}}\end{array}\right],$ then the matrix $\mathrm{A}^{31}$ is equal to
  • ✓
    $A^3$
  • B
    $A$
  • C
    $A^2$
  • D
    $I_3$
✓

Answer

Correct option: A.
$A^3$
a
$x^{2}+x+1=0$

$\alpha=\omega$

$\alpha^{2}=\omega^{2}$

$A=\frac{1}{\sqrt{3}}\left[\begin{array}{ccc}{1} & {1} & {1} \\ {1} & {\omega} & {\omega^{2}} \\ {1} & {\omega^{2}} & {\omega}\end{array}\right]$

$A^{2}=\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {0} & {1} \\ {0} & {1} & {0}\end{array}\right]$

$\Rightarrow \mathrm{A}^{4}=\mathrm{A}^{2} \cdot \mathrm{A}^{2}=\mathrm{I}_{3}$

$A^{31}=A^{28} . A^{3}=A^{3}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 3 and 4 . determinant and metrices→3 and 4 . determinant and metrices — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the principal values of: $\sin ^{-1}\left(\frac{-1}{2}\right)$
→
Let $f (1) = - 2$ and $f ' (x) \ge 4.2$ for $1 \le x \le 6$. The smallest possible value of $f (6)$, is
→
In a box containing 100 bulbs, 10 are defective. What is the probability that out of a sample of 5 bulbs, none is defective?

  1. $\big(\frac{9}{10}\big)^5$

  2. $\frac{9}{10}$

  3. $10^{-5}$

  4. $\big(\frac{1}{2}\big)^2$

→
$\int\frac{\sin^2\text{x}}{\cos^4\text{x}}\text{ dx}=$
  1. $\frac{1}{3}\tan^2\text{x}+\text{C}$
  2. $\frac{1}{2}\tan^2\text{x}+\text{C}$
  3. $\frac{1}{3}\tan^3\text{x}+\text{C}$
  4. none of these.
→
If $\alpha, \beta, \gamma$ are the direction angles of a vector and $\cos \alpha=\frac{14}{15}, \cos \beta=\frac{1}{3}$, then $\cos \gamma=$
→
For real numbers $\alpha, \beta, \gamma$ and $\delta,$ if  $\int \frac{\left(x^{2}-1\right)+\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)}{\left(x^{4}+3 x^{2}+1\right) \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)} d x$  $=\alpha \log _{e}\left(\tan ^{-1}\left(\frac{x^{2}+1}{x}\right)\right)$ $+\beta \tan ^{-1}\left(\frac{\gamma\left(x^{2}-1\right)}{x}\right)+\delta \tan ^{-1}\left(\frac{x^{2}+1}{x}\right)+C$  where $C$ is an arbitrary constant, then the value of $10(\alpha+\beta \gamma+\delta)$ is equal to ....... .
→
If $P(A)=\frac{1}{2}, P(B)=0,$ then $P(A | B)$ is
→
$\int_{ - 2}^2 {|1 - {x^2}|\,dx = } $
 
→
The solution of $\log \,\left( {\frac{{dy}}{{dx}}} \right) = ax + by$ is
→
Let $f, g: N \rightarrow N$ such that $f(n+1)=f(n)+f(1)$ $\forall \, n \in N$ and $g$ be any arbitrary function. Which of the following statements is $NOT$ true ?
→