Question
Check whether the following matrices are invertible or not. : $\left[\begin{array}{rr}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$
✓
Answer
Let $A=\left[\begin{array}{rr}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]$
Then, $|\mathrm{A}|=\left|\begin{array}{cc}\sec \theta & \tan \theta \\ \tan \theta & \sec \theta\end{array}\right|$
$= sec^2\theta – \tan^2\theta = 1 \neq 0.$
$\therefore $ A is a non-singular matrix.
Hence, $A^{-1}$ exist.
Then, $|\mathrm{A}|=\left|\begin{array}{cc}\sec \theta & \tan \theta \\ \tan \theta & \sec \theta\end{array}\right|$
$= sec^2\theta – \tan^2\theta = 1 \neq 0.$
$\therefore $ A is a non-singular matrix.
Hence, $A^{-1}$ exist.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Determine the order and degree of the following differential equations. state also whether they are linear or non linear.
$(\text{y"})^2+(\text{y})^3+\sin\text{y}=0$
→$(\text{y"})^2+(\text{y})^3+\sin\text{y}=0$
Rewrite the following statements without using if .. then.It 2 is a rational number then $\sqrt{2}$ is irrational number.
→State the truth Value of $x^2 = 25$
→Prove the theorem : The Cartesian equations of the line passing through $\mathrm{A}\left(x_1, y_1, z_1\right)$ and having direction ratios $a, b, c$ are $\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}$.
→If $A=\left[\begin{array}{cc}1 & 2 \\ 3 & -2 \\ -1 & 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 3 & 2 \\ 4 & -1 & 3\end{array}\right]$ then find the order of $A B$
→Write the negations of the following.Qudrilateral is a square if and only if it is a rhombus.
State if the following is not the probability mass function of a random variable. Give reasons for your answer.
| Y | −1 | 0 | 1 |
| P(Y) | 0.6 | 0.1 | 0.2 |
In the following switching circuit,: Write symbolic form
Write the order of the differential equation representing the famliy of curve $y = ax + a^3$.
→Write the negations of the following.$Ɐ n \in N, n^2 + n + 2$ is divisible by $4$.
→