Question

The Matrix $\begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix}$ is

• A. skew symmetric
• B. Symmetric
• C. symmetric if a=b
• D. Skew symmetric if b=c

Hint:

To identify a symmetric matrix at a glance, look at the elements mirrored across the main diagonal. If the elements at $(1,2)$ and $(2,1)$, $(1,3)$ and $(3,1)$, and $(2,3)$ and $(3,2)$ are identical, the matrix is symmetric.

Answer 1

The Correct Option is B

1. Definition of a Symmetric Matrix: A square matrix $A$ is said to be symmetric if it is equal to its transpose ($A = A^T$). In terms of elements, this means $a_{ij} = a_{ji}$ for all $i$ and $j$.

2. Performing the Transpose Operation: Let's find $A^T$ by swapping the rows and columns of $A$:

• Row 1 becomes Column 1: $[a \quad h \quad g]$

• Row 2 becomes Column 2: $[h \quad b \quad f]$

• Row 3 becomes Column 3: $[g \quad f \quad c]$
The resulting transpose matrix is: $$A^T = \begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix}$$

3. Comparison: Comparing $A$ and $A^T$: $$A = \begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix}, \quad A^T = \begin{bmatrix} a & h & g \\ h & b & f \\ g & f & c \end{bmatrix}$$ Since $A = A^T$, the matrix is fundamentally symmetric. Note that this property holds regardless of the specific values of $a, b,$ or $c$ on the main diagonal.