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
Question
Let $\beta$ be a real number. Consider the matrix $A=\left(\begin{array}{ccc}\beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2\end{array}\right)$
If $A^7-(\beta-1) A^6-\beta A^5$ is a singular matrix, then the value of $9 \beta$ is
✓

Answer

$A=\left(\begin{array}{ccc}\beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2\end{array}\right)|A|=-1$
$\Rightarrow\left|A^7-(\beta-1) A^6-\beta A^5\right|=0$
$\Rightarrow|A|^5\left|A^2-(\beta-1) A-\beta I\right|=0$
$\Rightarrow|A|^5\left|\left(A^2-\beta A\right)+A-\beta I\right|=0$
$\Rightarrow|A|^5|A(A-\beta I)+I(A-\beta I)|=0$
$|A|^5|(A+I)(A-\beta I)|=0$
$A+I=\left(\begin{array}{ccc}\beta+1 & 0 & 1 \\ 2 & 2 & -2 \\ 3 & 1 & -1\end{array}\right) $
$\Rightarrow|A+I|=-4,$
Here $|A| \neq 0 |A+I| \neq 0 $
$A-\beta I=\left(\begin{array}{ccc}0 & 0 & 1 \\ 2 & 1-\beta & -2 \\ 3 & 1 & -2-\beta\end{array}\right)$
$|A-\beta I|=2-3(1-\beta)=3 \beta-1=0 $
$\Rightarrow \beta=\frac{1}{3}$
$9 \beta=3$

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

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

Similar questions

Let $z$ be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum value of $|\mathrm{z}-(6+8 \mathrm{i})|$ is equal to :
→
$\left[\frac{2^{2020}+1}{2^{2018}+1}\right]+\left[\frac{3^{2020}+1}{3^{2018}+1}\right]+\left[\frac{4^{2020}+1}{4^{2018}+1}\right] +\left[\frac{5^{2020}+1}{5^{2018}+1}\right] + \left[\frac{6^{2020}+1}{6^{2018}+1}\right]$  is
→
Let $S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\right.$ $\left(\sin ^6 \theta+\cos ^6 \theta\right)=0$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3\left((\alpha-2)^2+(\beta-1)^2\right)$ equals....................
→
$150$ workers were engaged to finish a piece of work in a certain number of days. $4$ workers dropped the second day, $4$ more workers dropped the third day and so on. It takes eight more days to finish the work now. The number of days in which the work was completed is
→
Let $A + 2B =$ $\left[ {\begin{array}{*{20}{c}}1&2&0\\6&{ - 3}&3\\{ - 5}&3&1 \end{array}} \right]$ and $2A - B =$ $\left[ {\begin{array}{*{20}{c}}2&{ - 1}&5\\ 2&{ - 1}&6\\0&1&2\end{array}} \right]$ then $Tr (A) - Tr (B)$ has the value equal to
→
Let : $\overrightarrow{ a }=\hat{ i }+2 \hat{ j }+3 \hat{ k }, \overrightarrow{ b }=\hat{ i }-\hat{ j }+2 \hat{ k }$ and $\vec{c}=5 \hat{i}-3 \hat{j}+3 \hat{k}$ be there vectors. If $\vec{r}$ is a vector such that, $\overrightarrow{ r } \times \overrightarrow{ b }=\overrightarrow{ c } \times \overrightarrow{ b }$ and $\overrightarrow{ r } \cdot \overrightarrow{ a }=0$. Then $25|\overrightarrow{ r }|^2$ is equal to
→
Let $z_1$ and $z_2$ be two complex number such that $z_1$ $+z_2=5$ and $z_1^3+z_2^3=20+15 i$. Then $\left|z_1^4+z_2^4\right|$ equals-
→
If the first term of an $A.P. $ be $10$, last term is $50$ and the sum of all the terms is $300$, then the number of terms are
→
Let $\tan \alpha, \tan \beta$ and $\tan \gamma ; \alpha, \beta, \gamma \neq \frac{(2 n -1) \pi}{2}$ $n \in N$ be the slopes of three line segments $OA,OB$ and $OC$, respectively, where $O$ is origin.If circumcentre of $\Delta ABC$ coincides with origin and its orthocentre lies on $y-$axis, then the value of $\left(\frac{\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma}{\cos \alpha \cos \beta \cos \gamma}\right)^{2}$ is equal to :
→
The focus of the parabola $y^2=4 x+16$ is the centre of the circle C of radius 5 . If the values of $\lambda$, for which $C$ passes through the point of intersection of the lines $3 x-y=0$ and $x+\lambda y=4$, are $\lambda_1$ and $\lambda_2, \lambda_1<\lambda_2$, then $12 \lambda_1+29 \lambda_2$ is equal to __________.
→