Download App

MATRICES — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSMATRICES3 Marks
Question
If $\text{A}=\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix},$ verify that $A^TA = I_2.$
✓

Answer

Given: $\text{A}=\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix}$
$\text{A}^\text{T}=\begin{bmatrix}\sin\alpha&-\cos\alpha\\\cos\alpha&\sin\alpha\end{bmatrix}$
Now,
$\text{A}^\text{T}\text{A}=\begin{bmatrix}\sin\alpha&-\cos\alpha\\\cos\alpha&\sin\alpha\end{bmatrix}\begin{bmatrix}\sin\alpha&\cos\alpha\\-\cos\alpha&\sin\alpha\end{bmatrix}$
$\Rightarrow\text{A}^\text{T}\text{A}=[(\sin \alpha)(\sin \alpha)+(-\cos \alpha)(-\cos \alpha) \quad(\sin \alpha)(\cos \alpha)$$+(-\cos \alpha)(\sin \alpha) \ \cos \alpha)(\sin \alpha)$$+(\sin \alpha)(-\cos \alpha) \quad(\cos \alpha)(\cos \alpha)+(\sin \alpha)(\sin \alpha)]$
$\Rightarrow\text{A}^\text{T}\text{A}=\begin{bmatrix}\sin^2\alpha+\cos^2\alpha&\sin\alpha\cos\alpha-\sin\alpha\cos\alpha\\\sin\alpha\cos\alpha-\sin\alpha\cos\alpha&\cos^2\alpha+\sin^2\alpha\end{bmatrix}$
$\Rightarrow\text{A}^\text{T}\text{A}=\begin{bmatrix}1&0\\0&1\end{bmatrix}=\text{I}$

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 "3 Marks Question" 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

If  $\tan^{-1} \frac{\text{x - 3}}{\text{x - 4}} + \tan^{-1} \frac{\text{x + 3}}{\text{x + 4}} = \frac{\pi}{4},$ then find the value of x.
→
The volume of a spherical balloon is increasing at the rate of $25\ cm^3/ sec.$ Find the rate of change of its surface area at the instant when radius is $5\ cm.$
→
Evaluate the following definite integrals:
$\int_{0}^\limits{\pi}\frac{1}{1+\sin\text{x}}\text{ dx}$
→
Solve the following equation
$\text{xy dy}=(\text{y}-1)(\text{x}+1)\text{dx}$
→
Find the value of $\lambda$ so that the following vectors are coplanar:
$\vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}++\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}},\vec{\text{c}}=\lambda\hat{\text{i}}+\lambda\hat{\text{j}}+5\hat{\text{k}}$
→
Evaluate the following integrals:
$\int\frac{1-\cos\text{x}}{1+\cos\text{x}}\text{dx}$
→
Find f(x) is continuse at x = 0, then $\text{f(x)}=\frac{\text{x}}{1-\sqrt{1-\text{x}}}$ becomes continuous at x = 0.
→
Evaluate the following integrals:
$\int\frac{\cos\text{x}-\sin\text{x}}{1+\sin2\text{x}}\text{dx}$
→
Find the interval for which function $f(x)=2 \log (x-2)-x^2+4 x+1$ is increasing.
→
Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2cm?
→