If A = [ * 20 c 1&2&1 0&1& - 1 3& - 1 &1 ], then - Vidyadip

MCQ

If $A = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]$, then

A
${A^3} + 3{A^2} + A - 9{I_3} = O$
B
${A^3} - 3{A^2} + A + 9{I_3} = O$
C
${A^3} + 3{A^2} - A + 9{I_3} = O$
✓
${A^3} - 3{A^2} - A + 9{I_3} = O$

✓

Answer

Correct option: D.

${A^3} - 3{A^2} - A + 9{I_3} = O$

d
(d) $A = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]$

${A^2} = A\,.\,A = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]\,$$ = \left[ {\begin{array}{*{20}{c}}4&3&0\\{ - 3}&2&{ - 2}\\6&4&5\end{array}} \right]$

$A\,.\,{A^2} = \left[ {\begin{array}{*{20}{c}}1&2&1\\0&1&{ - 1}\\3&{ - 1}&1\end{array}} \right]\,\left[ {\begin{array}{*{20}{c}}4&3&0\\{ - 3}&2&{ - 2}\\6&4&5\end{array}} \right]\, = \left[ {\begin{array}{*{20}{c}}4&{11}&1\\{ - 9}&{ - 2}&{ - 7}\\{21}&{11}&7\end{array}} \right]$

==> ${A^3} - 3{A^2} - A + 9{I_3} = 0$.

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

Let $g ( x )=\int_{0}^{ x } f( t ) dt ,$ where $f$ is continuous function in $[0,3]$ such that $\frac{1}{3} \leq f(t) \leq 1$ for all $t \in[0,1]$ and $0 \leq f( t ) \leq \frac{1}{2}$ for all $t \in(1,3]$ The largest possible interval in which $g (3)$ lies is :

If ${I_1} = \int_0^1 {{2^{{x^2}}}dx,\;} {I_2} = \int_0^1 {{2^{{x^3}}}dx} ,\;{I_3} = \int_1^2 {{2^{{x^2}}}dx} $,${I_4} = \int_1^2 {{2^{{x^3}}}dx} $, then

Let $n \geq 2$ be an integer. Take $n$ distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of $n$ is

$\int_0^{\pi /2} {\frac{{\cos x - \sin x}}{{1 + \sin x\cos x}}} \,dx = $

Set $A$ has $m$ elements and Set $B$ has $n$ elements. If the total number of subsets of $A$ is $112$ more than the total number of subsets of $B$, then the value of $m \times n$ is

Let f, g: $R \rightarrow R$ be functions defined by $f ( x )=\left\{\begin{array}{ll}{[ x ]} & , \quad x < 0 \\ |1- x | & , \quad x \geq 0\end{array}\right.$ and $g(x)=\left\{\begin{array}{ll}e^{x}-x & , x < 0 \\ (x-1)^{2}-1 & , \quad x \geq 0\end{array}\right.$ where $[ x ]$ denote the greatest integer less than or equal to $x$. Then, the function fog is discontinuous at exactly

Area of the triangle formed by the lines $y^2 -9xy + 18x^2 = 0$ and $y = 9$ , is ............ $\mathrm{sq. \,unit}$

Coefficient of $x^{64}$ in the expansion of $(x - 1)^2(x - 2)^3(x - 3)^4(x - 4)^5 .... (x - 10)^{11}$

The direction cosines of a line segment $AB$ are $ - 2/\sqrt {17} ,$ $3/\sqrt {17} ,\,\, - 2/\sqrt {17} .$ If $AB = \sqrt {17} $ and the co-ordinates of $A$ are $(3, -6, 10)$, then the co-ordinates of $B$ are

If the area of the auxiliary circle of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\left( {a > b} \right)$ is twice the area of the ellipse, then the eccentricity of the ellipse is