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  • If matrix <img alt="A=\left[\begin{array}{cc}{a +bi} & {-c+i d} \\ { c+id} & {a -ib}\end{array}\right]" src="https://learn.careers360.com/latex-image/?A%3D%5Cleft%5B%5Cbegin%7Barray%7D

If matrix A=\left[\begin{array}{cc}{a +bi} & {-c+i d} \\ { c+id} & {a -ib}\end{array}\right] is Unitary matrix then

Option: 1

a^2+b^2+c^2+d^2=1


Option: 2

a^2+b^2=c^2+d^2


Option: 3

a^2+c^2=b^2+d^2


Option: 4

a^2+b^2+c^2+d^2=0


Answers (1)

best_answer

 

 

Orthogonal matrix, Unitary matrix and Idempotent matrix -

Unitary matrix

    Let A is a square matrix, and if AA?  = I, where I is the identity matrix, then A is said to be a unitary matrix.

Note:

    1. If AA?  =  I, then A-1 = A?

    2. If A and B are unitary, Then AB is also unitary.

    3. If A is unitary, then A-1 and A’ are also unitary.


 

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\\A=\left[\begin{array}{cc}{a +bi} & {-c+i d} \\ { c+id} & {a -ib}\end{array}\right]\\ \\A\cdot A^{\theta}=\left[\begin{array}{cc}{a +bi} & {-c+i d} \\ { c+id} & {a -ib}\end{array}\right]\cdot \left[\begin{array}{cc}{a -bi} & {c-i d} \\ { -c-id} & {a +ib}\end{array}\right]=\left[\begin{array}{cc}{1} & {0} \\ { 0} & {1}\end{array}\right]

\left[\begin{array}{cc}{a^2+b^2+c^2+d^2} & {0} \\ { 0} & {a^2+b^2+c^2+d^2}\end{array}\right]=\left[\begin{array}{cc}{1} & {0} \\ { 0} & {1}\end{array}\right]

a^2+b^2+c^2+d^2=1

hence, option (a) is correct

Posted by

vinayak

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