Question

Let $A$ be a square matrix and $A^{T}$ be its transpose. Which one of the following is true?

• A. $A+A^{T}$ is skew symmetric and $A-A^{T}$ is symmetric.
• B. both $A+A^{T}$ and $A-A^{T}$ are symmetric
• C. both $A+A^{T}$ and $A-A^{T}$ are skew symmetric
• D. $A+A^{T}$ is symmetric and $A-A^{T}$ is skew symmetric
• E. $-A-A^{T}$ is skew symmetric

Hint:

Math Tip: This is a standard theorem in matrix algebra. Any square matrix $A$ can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix: $A = \frac{1}{2}(A+A^T) + \frac{1}{2}(A-A^T)$.

Answer 1

The Correct Option is D

Concept:
Matrices - Properties of Symmetric and Skew-Symmetric Matrices.

• A matrix $M$ is symmetric if $M^T = M$.
• A matrix $M$ is skew-symmetric if $M^T = -M$.
• Transpose property: $(A + B)^T = A^T + B^T$ and $(A^T)^T = A$.

Step 1: Check the symmetry of $A + A^T$.
Let $P = A + A^T$. Take the transpose of $P$: $$ P^T = (A + A^T)^T $$ $$ P^T = A^T + (A^T)^T $$ $$ P^T = A^T + A $$ $$ P^T = A + A^T = P $$ Since $P^T = P$, the matrix $A + A^T$ is symmetric.
Step 2: Check the symmetry of $A - A^T$.
Let $Q = A - A^T$. Take the transpose of $Q$: $$ Q^T = (A - A^T)^T $$ $$ Q^T = A^T - (A^T)^T $$ $$ Q^T = A^T - A $$ Factor out a negative sign: $$ Q^T = -(A - A^T) = -Q $$ Since $Q^T = -Q$, the matrix $A - A^T$ is skew-symmetric.