Question

Which of the following statements are correct:
A. The product of two invertible matrices is invertible.
B. The product of two Hermitian matrices is Hermitian.
C. The product of two orthogonal matrices is orthogonal.
D. The product of two unitary matrices is unitary.
E. The product of two skew-symmetric matrices is symmetric.
Choose the correct answer from the options given below:

• A. A, C, D Only
• B. B, E Only
• C. A, B, E Only
• D. A, B, C, D Only

Hint:

The product of two Hermitian (or symmetric/skew-symmetric) matrices satisfies the same symmetry property if and only if the two matrices commute ($AB = BA$).

Answer 1

The Correct Option is A

Step 1: Concept:
This question asks for the algebraic properties of matrix products involving invertible, Hermitian, orthogonal, unitary, and skew-symmetric matrices.

Step 2: Key Formula or Approach:
Use matrix transpose ($T$), conjugate transpose ($\dagger$), and inverse properties:
1. Invertible: $(AB)^{-1} = B^{-1} A^{-1}$
2. Hermitian: $A^\dagger = A$, $B^\dagger = B$
3. Orthogonal: $A^T A = I$, $B^T B = I$
4. Unitary: $A^\dagger A = I$, $B^\dagger B = I$
5. Skew-symmetric: $A^T = -A$, $B^T = -B$

Step 3: Step-by-step Explanation:

• Statement A:
If $A$ and $B$ are invertible, $\det(AB) = \det(A)\det(B) \neq 0$, so $(AB)^{-1} = B^{-1}A^{-1}$ exists. Statement A is correct.

• Statement B:
If $A^\dagger = A$ and $B^\dagger = B$, then $(AB)^\dagger = B^\dagger A^\dagger = BA$.
$AB$ is Hermitian if and only if $AB = BA$ (i.e., $A$ and $B$ commute). In general, $BA \neq AB$, so Statement B is false.

• Statement C:
If $A^T A = I$ and $B^T B = I$, then: \[ (AB)^T (AB) = (B^T A^T)(A B) = B^T (A^T A) B = B^T I B = B^T B = I \] Thus, $AB$ is orthogonal. Statement C is correct.

• Statement D:
If $A^\dagger A = I$ and $B^\dagger B = I$, then: \[ (AB)^\dagger (AB) = (B^\dagger A^\dagger)(A B) = B^\dagger (A^\dagger A) B = B^\dagger I B = B^\dagger B = I \] Thus, $AB$ is unitary. Statement D is correct.

• Statement E:
If $A^T = -A$ and $B^T = -B$, then: \[ (AB)^T = B^T A^T = (-B)(-A) = BA \] For $AB$ to be symmetric, we require $(AB)^T = AB \implies BA = AB$. Since matrices do not generally commute, $AB$ is not necessarily symmetric. Statement E is false.

Step 4: Final Answer:
Statements A, C, and D are correct. Therefore, option (A) is the correct answer.