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
Let $S = \left\{ {\left( {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}\\
{{a_{21}}}&{{a_{22}}}
\end{array}} \right):{a_{ij}} \in \left\{ {0,1,2} \right\},{a_{11}} = {a_{22}}} \right\}$ Then the number of non-singular matrices in the set $S$ is
  • A
    $27$
  • B
    $24$
  • C
    $10$
  • ✓
    $20$
✓

Answer

Correct option: D.
$20$
d
The matrices in the form 

$\left[ {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}}\\
{{a_{21}}}&{{a_{22}}}
\end{array}} \right],{a_{ij}} \in \left\{ {0,1,2} \right\},{a_{  11}} = {a_{12}}$ are

$\left[ {\begin{array}{*{20}{c}}
0&{0/1/2}\\
{0/1/2}&0
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
1&{0/1/2}\\
{0/1/2}&1
\end{array}} \right].\left[ {\begin{array}{*{20}{c}}
2&{0/1/2}\\
{0/1/2}&2
\end{array}} \right]$

At any place, $0/1/2$ means $0,1$ or $2$ will be the element at that place.

Hence therefore total $\left( {27 = 3 \times 3 + 3 \times 3 + 3 \times 3} \right)$

matrices of the above form. Out of which the matrices which are singular are

$\left[ {\begin{array}{*{20}{c}}
0&{0/1/2}\\
0&0
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
0&0\\
{1/2}&0
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
1&1\\
1&1
\end{array}} \right],\left[ {\begin{array}{*{20}{c}}
2&2\\
2&2
\end{array}} \right]$

Hence these are total $7=(3+2+1+1)$ singular matrices.

Therefore number of all non-singular matrices in the given form $=27-7=20$

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

The approximate value of ${(7.995)^{1/3}}$ correct to four decimal places is
→
Let $M=\left[\begin{array}{cc}\sin ^4 \theta & -1-\sin ^2 \theta \\ 1+\cos ^2 \theta & \cos ^4 \theta\end{array}\right]=\alpha I +\beta M ^{-1}$, where $\alpha=\alpha(\theta)$ and $\beta=\beta(\theta)$ are real number, and $I$ is the $2 \times 2$ identity matrix. If $\alpha^*$ is the minimum of the set $\{\alpha(\theta): \theta \in[0,2 \pi)\}$ and $\beta^*$ is the minimum of the set $\{\beta(\theta): \theta \in[0,2 \pi)\}$, then the value of $\alpha^*+\beta^*$ is
→
$\left| {\frac{1}{2}({z_1} + {z_2}) + \sqrt {{z_1}{z_2}} } \right| + \left| {\frac{1}{2}({z_1} + {z_2}) - \sqrt {{z_1}{z_2}} } \right|$ =
→
If the area of the region $\left\{(x, y):-1 \leq x \leq 1,0 \leq y \leq a+e^{|x|}-e^{-x}, a>0\right\}$ is $\frac{ e ^2+8 e +1}{ e }$, then the value of a is :
→
The area of the region  $S=\left\{(x, y): 3 x^{2} \leq 4 y \leq 6 x+24\right\} \text { is }...... \,.$
→
In how many ways can $10$ balls be divided between two boys, one receiving two and the other eight balls
→
Let $\alpha $ and $\beta $ be the roots of the equation $x^2 + x + 1 = 0.$ Then for $y \ne 0$ in $R,$ $\left| {\begin{array}{*{20}{c}}
{y\, + \,1}&\alpha &\beta \\
\alpha &{y\, + \,\beta }&1\\
\beta &1&{y\, + \,\alpha }
\end{array}} \right|$ is equal to
→
If $a\,.\,b = b\,.\,c = c\,.\,a = 0$ , then what is the value of the scalar triple product $ [a b c] $ ?
→
The nth derivative of $x{e^x}$ vanishes when
→
The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $\mathrm{x}^{2}=4 \mathrm{y},$ internally in the ratio $1: 2,$ is 
→