Download App

STD 12 - 3 and 4 . determinant and metrices — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 12 - 3 and 4 . determinant and metrices4 Marks
MCQ
If $A = \left[ {\begin{array}{*{20}{c}}a&0&0\\0&b&0\\0&0&c\end{array}} \right]$, then ${A^n} = $
  • A
    $\left[ {\begin{array}{*{20}{c}}{na}&0&0\\0&{nb}&0\\0&0&{nc}\end{array}} \right]$
  • B
    $\left[ {\begin{array}{*{20}{c}}a&0&0\\0&b&0\\0&0&c\end{array}} \right]$
  • ✓
    $\left[ {\begin{array}{*{20}{c}}{{a^n}}&0&0\\0&{{b^n}}&0\\0&0&{{c^n}}\end{array}} \right]$
  • D
    None of these
✓

Answer

Correct option: C.
$\left[ {\begin{array}{*{20}{c}}{{a^n}}&0&0\\0&{{b^n}}&0\\0&0&{{c^n}}\end{array}} \right]$
c
(c) Since ${A^2} = A.A = \left[ {\begin{array}{*{20}{c}}a&0&0\\0&b&0\\0&0&c\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}a&0&0\\0&b&0\\0&0&c\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}{{a^2}}&0&0\\0&{{b^2}}&0\\0&0&{{c^2}}\end{array}} \right]$

And ${A^3} = \left[ {\begin{array}{*{20}{c}}{{a^3}}&0&0\\0&{{b^3}}&0\\0&0&{{c^3}}\end{array}} \right]$,....

==> ${A^n} = {A^{n - 1}}.A = \left[ {\begin{array}{*{20}{c}}{{a^{n - 1}}}&0&0\\0&{{b^{n - 1}}}&0\\0&0&{{c^{n - 1}}}\end{array}} \right]{\rm{ }}\left[ {\begin{array}{*{20}{c}}a&0&0\\0&b&0\\0&0&c\end{array}} \right]$

$ = \left[ {\begin{array}{*{20}{c}}{{a^n}}&0&0\\0&{{b^n}}&0\\0&0&{{c^n}}\end{array}} \right]$.

Note: Students should remember this question as a formula.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

$\int_{}^{} {\frac{{dx}}{{(x - {x^2})}} = } $
→
If $x + iy =\sqrt {\phi  + i\psi } \,$ where $  i = \sqrt { - 1} \,$ and $\phi$  and $\psi$  are non zero real parameters then $ \phi =$  constant and $ \psi  = $ constant, represents two systems of rectangular hyperbola which intersect at an angle of
→
Let a circle passing through $(2,0)$ have its centre at the point $( h , k )$. Let $\left( x _{ c }, y _{ c }\right)$ be the point of intersection of the lines $3 x+5 y=1$ and $(2+c) x+$ $5 c^2 y=1$. If $h=\lim _{c \rightarrow 1} x_c$ and $k=\lim _{c \rightarrow 1} y_c$, then the equation of the circle is $:$
→
For $0< a <1,$ the value of the integral $\int_0^\pi \frac{d x}{1-2 a \cos x+a^2}$ is :
→
The number of distinct primes dividing $12 !+13 !+14 !$ is
→
If $A$ and $B$ are two invertible square matrices of the same order such that $(A + B)(A -B) = A^2-B^2$, then $(A^2BA^{-1}B^{-1})^3$ is equal to-
→
The area bounded by the parabola ${y^2} = 4ax,$ its axis and two ordinates $x = 4,$ $x = 9$ is
→
If $x$ co-ordinates of a point $P$ of line joining the points $Q(2,\,2,\,1)$ and $R\,(5,\,2, - 2)$ is $4$, then the $z$ - coordinates of $P$ is
→
$\int_{\,0}^{\,1000} {{e^{x - [x]}}dx} $ is
→
If ${\left( {\frac{{1 + \cos \theta + i\sin \theta }}{{i + \sin \theta + i\cos \theta }}} \right)^4} = \cos n\theta + i\sin n\theta $, then $n$ is equal to
→