Download App

Matrices — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMatrices5 Marks
Question
If $A=\left[\begin{array}{cc}0 & -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} & 0\end{array}\right]$ and I is the identity matrix of order 2 , show that
$I+A=(I-A)\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$
✓

Answer

L.H.S. $I + A = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&{ - \tan \frac{\alpha }{2}} \\ {\tan \frac{\alpha }{2}}&0 \end{array}} \right]$$ = \left[ {\begin{array}{*{20}{c}} 1&{ - \tan \frac{\alpha }{2}} \\ {\tan \frac{\alpha }{2}}&1 \end{array}} \right]$
Now, $I - A = \left[ {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 0&{ - \tan \frac{\alpha }{2}} \\ {\tan \frac{\alpha }{2}}&0 \end{array}} \right]$$ = \left[ {\begin{array}{*{20}{c}} 1&{\tan \frac{\alpha }{2}} \\ { - \tan \frac{\alpha }{2}}&1 \end{array}} \right]$
R.H.S. $ = (I - A)\left[ {\begin{array}{*{20}{c}} {\cos \alpha }&{ - \sin \alpha } \\ {\sin \alpha }&{\cos \alpha } \end{array}} \right]$ $ = \left[ {\begin{array}{*{20}{c}} 1&{\tan \frac{\alpha }{2}} \\ { - \tan \frac{\alpha }{2}}&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\cos \alpha }&{ - \sin \alpha } \\ {\sin \alpha }&{\cos \alpha } \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} {\cos \alpha + \sin \alpha \tan \frac{\alpha }{2}}&{ - \sin \alpha + \cos \alpha \tan \frac{\alpha }{2}} \\ { - \cos \alpha \tan \frac{\alpha }{2} + \sin \alpha }&{\sin \alpha \tan \frac{\alpha }{2} + \cos \alpha } \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} {\cos \alpha + \sin \alpha \frac{{\sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}}&{ - \sin \alpha + \cos \alpha \frac{{\sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}} \\ { - \cos \alpha \frac{{\sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}} + \sin \alpha }&{\sin \alpha \frac{{\sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}} + \cos \alpha } \end{array}} \right]$
$=\left[ {\begin{array}{*{20}{c}} {\frac{{\cos \alpha \cos \frac{\alpha }{2} + \sin \alpha \sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}}&{\frac{{ - \sin \alpha \cos \frac{\alpha }{2} + \cos \alpha \sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}} \\ {\frac{{ - \cos \alpha \sin \frac{\alpha }{2} + \sin \alpha \cos \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}}&{\frac{{\sin \alpha \sin \frac{\alpha }{2} + \cos \alpha \cos \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}} \end{array}} \right]$
$ = \left[ {\begin{array}{*{20}{c}} {\frac{{\cos \left( {\alpha - \frac{\alpha }{2}} \right)}}{{\cos \frac{\alpha }{2}}}}&{\frac{{ - \sin \left( {\alpha - \frac{\alpha }{2}} \right)}}{{\cos \frac{\alpha }{2}}}} \\ {\frac{{\sin \left( {\alpha - \frac{\alpha }{2}} \right)}}{{\cos \frac{\alpha }{2}}}}&{\frac{{\cos \left( {\alpha - \frac{\alpha }{2}} \right)}}{{\cos \frac{\alpha }{2}}}} \end{array}} \right]$$ = \left[ {\begin{array}{*{20}{c}} {\frac{{\cos \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}}&{\frac{{ - \sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}} \\ {\frac{{\sin \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}}&{\frac{{\cos \frac{\alpha }{2}}}{{\cos \frac{\alpha }{2}}}} \end{array}} \right]$$ = \left[ {\begin{array}{*{20}{c}} 1&{ - \tan \frac{\alpha }{2}} \\ {\tan \frac{\alpha }{2}}&1 \end{array}} \right]$
$\therefore$ L.H.S. = R.H.S. Proved.

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 "5 Marks Questions" from Matrices→Matrices — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following definite integrals:
$\int\limits_{\text{e}}^{\text{e}^2}\Big\{\frac{1}{\log\text{x}}-\frac{1}{(\log\text{x})^2}\Big\}\text{dx}$
→
$\int\frac{1}{\text{x}^{\frac{1}{3}}\big(\text{x}^{\frac{1}{3}}-1\big)}\text{dx}$
→
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = a - b for all a, b ∈ Q.
→
Find the mean number of heads in three tosses of a fair coin.
→
Verify Rolle's theorem of the following function on the indicated interval
$\text{f}(\text{x})=\text{e}^{\text{x}}\cos{\text{x}}\text{ on }\Big[\frac{-\pi}{2},\frac{\pi}{2}\Big]$
→
Find whether the function is differentiable at x = 1 and x = 2
$\text{f(x)}=\begin{cases}\text{x} & \text{x}\leq1\\2-\text{x} & 1\leq\text{x}\leq2\\-2+3\text{x}&\text{x}>2\end{cases}$
→
Show that the points A(1, -2, -8), B(5, 0, -2) and C(11, 3, 7) are collinear, and find the ratio in which B divides AC.
→
Express the vector $\vec{\text{a}}=5\hat{\text{i}}-2\hat{\text{j}}+5\hat{\text{k}}$ as the sum of two vectors such that one is parallel to the vector $\vec{\text{b}}=3\hat{\text{i}}+\hat{\text{k}}$ and other is perpendicular to $\vec{\text{b}}.$
→
Evaluate the following definite integrals:
$\int\limits_{0}^{\frac{\pi}{2}}\sin^3\text{x}\text{ dx}$
→
A and B are two independent events. The probability that A and B occur is $\frac{1}{6}$ and the probability that neither of them occurs is $\frac{1}{3}$. Find the probability of occurrence of two events.
→