Download App

Algebra of Matrices — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsAlgebra of Matrices3 Marks
Question
If $\text{A}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix},$ then verify that $A^TA = I_2$.
✓

Answer

$\text{A}=\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}$
$\therefore\ \text{A}'=\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}$ $\text{A}'\text{A}=\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}\begin{bmatrix}\cos\alpha&\sin\alpha\\-\sin\alpha&\cos\alpha\end{bmatrix}$
$\begin{bmatrix}(\cos\alpha)(\cos\alpha)+(-\sin\alpha)(-\sin\alpha)&(\cos\alpha)(\sin\alpha)+(-\sin\alpha)(\cos\alpha)\sin\alpha)(\cos\alpha)+(\cos\alpha)(-\sin\alpha)&(\sin\alpha)(\sin\alpha)+(\cos\alpha)(\cos\alpha)\end{bmatrix}$
$\begin{bmatrix}\cos^2\alpha+\sin^2\alpha&\sin\alpha\cos\alpha-\sin\alpha\cos\alpha\\\sin\alpha\cos\alpha-\sin\alpha\cos\alpha&\sin^2\alpha+\cos^2 \alpha\end{bmatrix}$ $=\begin{bmatrix}1&0\\0&1\end{bmatrix}=\text{I}$
Hence, we have verified that A'A = 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 "Solve the Following Question.(3 Marks)" from Algebra of Matrices→Algebra of Matrices — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If A = {1, 2, 3}, show that a onto function f : A → A must be one-one.
→
Find the vector equation of a plane which is at 42 unit distance from the origin and which

is normal to the vector $2 \hat{i}+\hat{j}-2 \hat{k}$

→
Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = 2x^2 - 3x + 1$ on $[1, 3]$
→
Show that $\cot ^{-1} \frac{1}{3}-\tan ^{-1} \frac{1}{3}=\cot ^{-1} \frac{3}{4}$.
→
A coin is tossed three times. Find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ in each of the following:
A = At least two heads,
B = At most two heads.
→
If $\text{f(x)}=\frac{2\text{x}+3\sin\text{x}}{3\text{x}+2\sin\text{x}},\text{ x}\neq0$ is continuous at x = 0, then find f(0).
→
Write the vector equation of the line passing through the point (1, -2, -3) and normal to the plane $\vec{\text{r}}.(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}})=5.$
→
Evalute the following integrals:
$\int\frac{\sec^2\text{x}}{\tan\text{x}+2}\text{dx}$
→
Write the derivative of $\sin\text{x}$ with respect to $\cos\text{x}$.
→
The volume of the spherical ball is increasing at the rate of $4 \pi \mathrm{cc} / \mathrm{sec}$. Find the rate at which the radius and the surface area are changing when the volume is $288 \pi \mathrm{cc}$.
→