Question

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

Answer 1

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.

Complete step-by-step answer:
In mathematics, a complex square matrix A is unitary if its conjugate transpose $$A^{\ast }$$is also its inverse.
A unitary matrix can be defined as a square complex matrix A for which,
 $$A{A}^{\ast }=A^{\ast }A=I$$
$$A^{\ast }$$= 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.
$$A{A}^{\ast }=A^{\ast }A=I$$
Taking determinant of complete equation.
$$\Rightarrow \left|A{A}^{\ast }\right|=\left|A^{\ast }A\right|=\left|I\right|$$
Separating the determinant of each term in the equation.
$$\Rightarrow \left|\left|A\right|\times \left|A^{\ast }\right|\right|=\left|\left|A^{\ast }\right|\times \left|A\right|\right|=\left|I\right|$$
Removing the determinant above the whole equation of both sides.
$$\Rightarrow \left|A\right|\times \left|A^{\ast }\right|=\left|A^{\ast }\right|\times \left|A\right|=1$$
Now cancelling$$\left|A^{\ast }\right|$$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 $$A{A}^{\ast }=A^{\ast }A=I$$ where $$A^{\ast }$$= 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 $$A^{\ast }$$ 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 $$A^{-1}=A^{\ast }$$
5) If A is Unitary matrix then it's determinant is of Modulus Unity (always1).