Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# If $A$ is an unitary matrix then $\left| A \right|$ is equal to:A) $1$B) $- 1$C) $\pm 1$D) $2$

Last updated date: 16th Jun 2024
Total views: 402.6k
Views today: 7.02k
Verified
402.6k+ views
Hint: 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.

In mathematics, a complex square matrix A is unitary if its conjugate transpose ${A^ * }$is also its inverse.
A unitary matrix can be defined as a square complex matrix A for which,
$A{A^*} = {A^*}A = I$
${A^*}$= Conjugate transpose of A
$I$= 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.
$A{A^*} = {A^*}A = I$
Taking determinant of complete equation.
$\Rightarrow \left| {A{A^*}} \right| = \left| {{A^*}A} \right| = \left| I \right|$
Separating the determinant of each term in the equation.
$\Rightarrow \left| {\left| A \right| \times \left| {{A^*}} \right|} \right| = \left| {\left| {{A^*}} \right| \times \left| A \right|} \right| = \left| I \right|$
Removing the determinant above the whole equation of both sides.
$\Rightarrow \left| A \right| \times \left| {{A^*}} \right| = \left| {{A^*}} \right| \times \left| A \right| = 1$
Now cancelling$\left| {{A^*}} \right|$from the equation we get,
$\Rightarrow \left| A \right| = \left| A \right| = 1$
$|A|$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 $A{A^*} = {A^*}A = I$ where ${A^*}$= Transpose Conjugate of A = ${\left( {A\prime } \right)^T}$ (first you Conjugate and then Transpose , you will get Unitary matrix)
Properties of Unitary matrix:
1) If A is a Unitary matrix then${A^{ - 1}}$is also a Unitary matrix.
2) If A is a Unitary matrix then ${A^*}$ 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 ${A^{ - 1}} = {A^*}$
5) If A is Unitary matrix then it's determinant is of Modulus Unity (always1).