Check if the following matrix is unitary- - 1-21-i 1-2-1-i- 1-21 - i 1-21 - i -

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard IX

Mathematics

Multiplication of Matrices

Check if the ...

Question

Check if the following matrix is unitary:

$⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} (1 + i) & \frac{1}{2} (- 1 + i) \frac{1}{2} (1 + i) & \frac{1}{2} (1 - i) \end{matrix} ⎤ ⎥ ⎥ ⎦$

Open in App

Solution

A matrix $U$ is said to be unitary matrix if $U U^{} = U^{} U = I$
where $U^{}$ is the conjugate transpose of $U$ and $I$ is an identity matrix.

Let $U = ⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} (1 + i) & \frac{1}{2} (- 1 + i) \frac{1}{2} (1 + i) & \frac{1}{2} (1 - i) \end{matrix} ⎤ ⎥ ⎥ ⎦$

$U^{} = ⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} (1 - i) & \frac{1}{2} (1 - i) \frac{1}{2} (- 1 - i) & \frac{1}{2} (1 + i) \end{matrix} ⎤ ⎥ ⎥ ⎦$

$U U^{*} = ⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} (1 + i) & \frac{1}{2} (- 1 + i) \frac{1}{2} (1 + i) & \frac{1}{2} (1 - i) \end{matrix} ⎤ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} (1 - i) & \frac{1}{2} (1 - i) \frac{1}{2} (- 1 - i) & \frac{1}{2} (1 + i) \end{matrix} ⎤ ⎥ ⎥ ⎦ = [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]$

$∴ U$ is an unitary matrix.

Suggest Corrections

Similar questions

Q. Express the following complex numbers in the standard form a + i b:
(i)

(1 + i) (1 + 2 i)

(ii)

\frac{3 + 2 i}{- 2 + i}

(iii)

\frac{1}{(2 + i)^{2}}

(iv)

\frac{1 - i}{1 + i}

(v)

\frac{(2 + i)^{3}}{2 + 3 i}

(vi)

\frac{(1 + i) (1 + \sqrt{3} i)}{1 - i}

(vii)

\frac{2 + 3 i}{4 + 5 i}

(viii)

\frac{(1 - i)^{3}}{1 - i^{3}}

(ix)

(1 + 2 i)^{- 3}

(x)

\frac{3 - 4 i}{(4 - 2 i) (1 + i)}

(xi)

(\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{1 - 4 i}{5 + i})

(xii)

\frac{5 + \sqrt{2} i}{1 - 2 \sqrt{i}}

Q. One value of

(1 + i)^{1 / 2}

2^{1 / 4} e^{i π / 8}

. The other value is

Express the following complex numbers in the standard form a + i b :

$(i) (1 + i) (1 + 2 i) (i i) \frac{3 + 2 i}{- 2 + i} (i i i) \frac{1}{(2 + i)^{2}} (i v) \frac{1 - i}{1 + i} (v) \frac{(2 + i)^{3}}{2 + 3 i} (v i) \frac{(1 + i) (1 + \sqrt{3 i})}{1 - i} (v i i) \frac{2 + 3 i}{4 + 5 i} (v i i i) \frac{(1 - i)^{3}}{1 - i^{3}} (i x) (1 + 2 i)^{- 3} (x) \frac{3 - 4 i}{(4 - 2 i) (1 + i)} (x i) (\frac{1}{1 - 4 i} - \frac{2}{1 + i}) (\frac{3 - 4 i}{5 + i}) (x i i) \frac{5 + \sqrt{2 i}}{1 - \sqrt{2 i}}$

Q. Assertion :If

z_{0} = \frac{1}{2} (1 + i)

, then

P_{n} (z) = (1 + z_{0}) (1 + {z_{0}}^{2}) (1 + {z_{0}}^{2^{2}}) . . . (1 + {z_{0}}^{2^{n}}) = (1 + i) (1 - \frac{1}{2^{2^{n}}})

, where

n > 1

is a positive integer. Reason:

P_{n} (z) = \frac{1 - {z_{0}}^{2^{n + 1}}}{1 - z_{0}}

Q. If

I_{1} = \int_{0}^{\frac{π}{2}} \frac{{cos}^{2} x}{1 + {cos}^{2} x} d x,

I_{2} = \int_{0}^{\frac{π}{2}} \frac{{sin}^{2} x}{1 + {sin}^{2} x} d x,

I_{3} = \int_{0}^{\frac{π}{2}} \frac{1 + 2 {cos}^{2} x {sin}^{2} x}{4 + 2 {cos}^{2} x {sin}^{2} x} d x

, then

Join BYJU'S Learning Program

Check if the following matrix is unitary:⎡⎢ ⎢⎣12(1+i)12(−1+i)12(1+i)12(1−i)⎤⎥ ⎥⎦

Check if the following matrix is unitary:

$⎡ ⎢ ⎢ ⎣ \begin{matrix} \frac{1}{2} (1 + i) & \frac{1}{2} (- 1 + i) \frac{1}{2} (1 + i) & \frac{1}{2} (1 - i) \end{matrix} ⎤ ⎥ ⎥ ⎦$