Question

Let $A$ be a symmetric matrix, then

• A. $(kA)^T = -kA$.
• B. $A^m$ is a symmetric matrix if $m$ is a positive integer.
• C. $A^T = -A$.
• D. $A + B$ is a symmetric matrix if $B$ is a skew symmetric matrix.

Hint:

Powers of symmetric matrices are always symmetric: $(A^m)^T = (A^T)^m = A^m$. Powers of skew-symmetric matrices are symmetric for even powers and skew-symmetric for odd powers.

Answer 1

The Correct Option is B

Step 1: Concept
A square matrix $A$ is defined to be symmetric if it is equal to its transpose, i.e., $A^T = A$.
A matrix $B$ is defined to be skew-symmetric if $B^T = -B$.

Step 2: Key Formulas and Approach
We test the fundamental matrix transpose properties: 1. $(A^T)^T = A$ 2. $(A + B)^T = A^T + B^T$ 3. $(kA)^T = k A^T$ 4. $(A_1 A_2 \dots A_m)^T = A_m^T \dots A_2^T A_1^T$ 5. $(A^m)^T = (A^T)^m$ for any positive integer $m$.

Step 3: Step-by-step Explanation

• Testing Option (A):
$(kA)^T = k A^T$. Since $A^T = A$, $(kA)^T = k A$. Thus $(kA)^T = -kA$ is false (unless $k=0$ or $A=O$).

• Testing Option (B):
Consider $(A^m)^T$. Using exponent-transpose property: \[ (A^m)^T = (A^T)^m \] Since $A$ is symmetric, $A^T = A$. Substituting $A^T = A$: \[ (A^m)^T = A^m \] Since $(A^m)^T = A^m$, $A^m$ is indeed a symmetric matrix for any positive integer $m$. Thus, Option (B) is true.

• Testing Option (C):
$A^T = -A$ is the defining equation of a skew-symmetric matrix, not a symmetric matrix. Thus, Option (C) is false.

• Testing Option (D):
Let $B$ be skew-symmetric, so $B^T = -B$. Then: \[ (A + B)^T = A^T + B^T = A - B \] Since $A - B \neq A + B$ in general, $A + B$ is not symmetric. Thus, Option (D) is false.

Step 4: Final Answer
If $A$ is symmetric, $A^m$ is also symmetric for every positive integer $m$. Thus, Option (B) is correct.