Solveeit Logo
Login

Question

Question: If \[A\] is an unitary matrix then \[\left| A \right|\] is equal to: A) \(1\) B) \( - 1\) C) \...

If AA is an unitary matrix then ∣A∣\left| A \right| is equal to:
A) 11
B) −1 - 1
C) ±1 \pm 1
D) 22

Explanation

Solution

A square matrix A is said to be unitary if its transpose is its own inverse and all its entries should belong to complex numbers. A unitary matrix is a matrix whose inverse equals its conjugate transpose. Unitary matrices are the complex analog of real orthogonal matrices.

Complete step-by-step answer:
In mathematics, a complex square matrix A is unitary if its conjugate transpose A∗{A^ * }is also its inverse.
A unitary matrix can be defined as a square complex matrix A for which,
AA∗=A∗A=IA{A^*} = {A^*}A = I
A∗{A^*}= Conjugate transpose of A
II= Identity matrix
When we are working with square matrices we are mapping a finite dimensional space to itself whenever we multiply.
Now let's take a situation where we are finding the determinant of the complete equation mentioned above.
AA∗=A∗A=IA{A^*} = {A^*}A = I
Taking determinant of complete equation.
⇒∣AA∗∣=∣A∗A∣=∣I∣\Rightarrow \left| {A{A^*}} \right| = \left| {{A^*}A} \right| = \left| I \right|
Separating the determinant of each term in the equation.
⇒∣∣A∣×∣A∗∣∣=∣∣A∗∣×∣A∣∣=∣I∣\Rightarrow \left| {\left| A \right| \times \left| {{A^*}} \right|} \right| = \left| {\left| {{A^*}} \right| \times \left| A \right|} \right| = \left| I \right|
Removing the determinant above the whole equation of both sides.
⇒∣A∣×∣A∗∣=∣A∗∣×∣A∣=1\Rightarrow \left| A \right| \times \left| {{A^*}} \right| = \left| {{A^*}} \right| \times \left| A \right| = 1
Now cancelling∣A∗∣\left| {{A^*}} \right|from the equation we get,
⇒∣A∣=∣A∣=1\Rightarrow \left| A \right| = \left| A \right| = 1
∣A∣|A|can be a complex number with modulus/magnitude 1.

So, option (A) is the correct answer.

Note: If matrix A is called Unitary matrix then it satisfy this condition AA∗=A∗A=IA{A^*} = {A^*}A = I where A∗{A^*}= Transpose Conjugate of A = (A′)T{\left( {A\prime } \right)^T} (first you Conjugate and then Transpose , you will get Unitary matrix)
Properties of Unitary matrix:

  1. If A is a Unitary matrix thenA−1{A^{ - 1}}is also a Unitary matrix.
  2. If A is a Unitary matrix then A∗{A^*} is also a Unitary matrix.
  3. If A&B; are Unitary matrices, then A.B is a Unitary matrix.
  4. If A is Unitary matrix then A−1=A∗{A^{ - 1}} = {A^*}
  5. If A is Unitary matrix then it's determinant is of Modulus Unity (always1).