Find the characteristic roots of the two-rowed orthogonal matrix `[(cos theta,-sin theta),(sin theta,cos theta)]` and verify that they are of unit mod

Monjur Alahi

Find the characteristic roots of the two-rowed orthogonal matrix `[(cos theta,-sin theta),(sin theta,cos theta)]` and verify that they are of unit modulus.


0 like 0 dislike
1 views
Share with your friends

1 Answers

Salim Ahmed Kawsar

Call

We have,
`|A-lambdaI|=|(cos theta - lambda,-sin theta),(sin theta,cos theta - lambda)|`
`=(cos theta-lambda)^(2)+sin^(2) theta`
Therefore, characteristic equation of A is
`(cos theta-lambda)^(2)+sin^(2) theta=0`
or `cos theta-lambda= pm i sin theta`
or `lambda=cos theta pm i sin theta`
Which are of unit modulus.

0 like 0 dislike
1 views
Talk Doctor Online in Bissoy App
Let `A=[{:(,sin theta,1sqrt2),(,-1//sqrt2,cos theta),(,cos theta,tan theta):}]& B=[{:(,1sqrt2,sin theta),(,cos theta,cos theta),(,cos theta,-1):}]`. F

By defing A & B are equal if they have the same order and all the corresponding elements are equal. Thus we have `sin theta=(1)/(sqrt2),c os theta=-(1)/(sqrt2)& tan theta=-1` `Rightarrow...

2 Answers 1 views
Prove the orthogonal matrices of order two are of the form `[(cos theta,-sin theta),(sin theta,cos theta)]` or `[(cos theta,sin theta),(sin theta,-cos

Let matrix `A=[(a,b),(c,d)]` is orthogonal matrix. `:. [(a,b),(c,d)][(a,c),(b,d)]=[(1,0),(0,1)]` `implies [(a^(2)+b^(2),ac+bd),(ac+bd,c^(2)+d^(2))]=[(1,0),(0,1)]` `implies a^(2)+b^(2)=1` (1) `c^(2)+d^(2)=1` (2) `ac+bd=0` (3) `implies a/d= (-b)/c= k` (let) `implies c^(2)+d^(2)=(a^(2)+b^(2))/k=1//k^(2)` or `k^(2)=1` or `k= pm 1`...

2 Answers 1 views
If `A` and `B` are matrices of the same order, then `A B^T-B^T A` is a (a) skew-symmetric matrix (b) null matrix (c) unit matrix (d) symmetric matrix

Let `P=(AB^(T)-BA^(T))` `:. P^(T)=(AB^(T)-BA^(T))^(T)=(AB^(T))^(T)-(BA^(T))^(T)` `=(B^(T))^(T) (A)^(T)-(A^(T))^(T)B^(T)=BA^(T)-AB^(T)` `=-(AB^(T)-BA^(T))=-P` Hence, `(AB^(T)-BA^(T))` is a skew-symmetric matric.

2 Answers 1 views
If `theta-phi=pi/2,` prove that, `[(cos^2 theta,cos theta sin theta),(cos theta sin theta,sin^2 theta)] [(cos^2 phi,cos phi sin phi),(cos phi sin phi,

Correct Answer - C `AB=[(cos^(2) theta,),(cos theta sin theta,)][(cos^(2) phi,cos phi sin phi),(cos phi sin phi,sin^(2) phi)]` `=[(cos^(2) theta cos^(2) phi+cos theta cos phi sin theta sin phi ,cos^(2)theta cos phi...

2 Answers 1 views
If `[alphabetagamma-alpha]` is to be square root of two-rowed unit matrix, then `alpha,betaa n dgamma` should satisfy the relation. `1-alpha^2+betagam

Correct Answer - B We have `[(alpha,beta),(gamma,-alpha)][(alpha,beta),(gamma,-alpha)]=[(1,0),(0,1)]` or `[(alpha^(2)+beta gamma,0),(0,alpha^(2)+beta gamma)]=[(1,0),(0,1)]` or `alpha^(2)+beta gamma-1=0`

2 Answers 5 views
If `A(theta)=[(sin theta, i cos theta),(i cos theta, sin theta)]`, then which of the following is not true ?

Correct Answer - A::B::C We have, `|A(theta)|=1` Hence, A is invertiable. `A(pi+theta)=A(pi + theta)=[(sin(pi+theta),i cos (pi+theta)),(i cos (pi+theta),sin (pi+theta))]` `=[(-sin theta,-i co theta),(-i cos theta,-sin theta)]=-A(theta)` adj `(A(theta))=[(sin theta,-i cos theta),(-i...

2 Answers 1 views
Find the principal values of the following (i) `sin^(-1) (sin 1)` (ii) `sin^(-1) (sin 2)` (iii) `sin^(-1) (sin 10)` (iv) `sin^(-1) (sin 20)` (v) `sin^

(i) Since `1 in [-pi//2, pi//2], sin^(-1) (sin 1) = 1` (ii) Since `2 in [-pi//2, pi//2], sin^(-1) (sin 2) != 2` `:. Sin^(-1) (sin 2) = sin^(-1) (sin (pi)...

2 Answers 1 views
Find the principal value of the following (i) `cos^(-1) (cos 3)` (ii) `cos^(-1) (cos 4)` (iii) `cos^(-1) (cos 15)` (iv) `cos^(-1) (cos 30)` (v) `cos^(

(i) Since `3 in [0, pi], cos^(-1) (cos 3) = 3` (ii) Since `4 !in [0, pi], cos^(-1) (cos 4) != 4` `:. Cos^(-1) (cos 4) = 2pi - 4`...

2 Answers 1 views
If equation `sin^(-1) (4 sin^(20 theta + sin theta) + cos^(-1) (6 sin theta - 1) = (pi)/(2)` has 10 solution for `theta in [0, n pi]`, then find the m

Correct Answer - 9 We must have `4 sin^(2) theta + sin theta = 6 sin theta -1` `rArr 4 sin^(2) theta - 5 sin theta + 1 = 0` `rArr...

2 Answers 1 views
सरल कीजिए, ` cos theta [{:( cos theta,, sin theta ), (-sin theta,, cos theta ):}] + sin theta [ {:(sin theta ,, - cos theta ), (cos theta ,, sin theta

Correct Answer - `[{:(,1,0),(,1,1):}]`

2 Answers 1 views
X