Q.If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.

The key idea is that a matrix is symmetric if it equals its own transpose, and the transpose of a sum is the sum of the transposes.

Let $A$, $B$, and $C$ be symmetric matrices of the same order. By definition:

Consider their sum $S = A + B + C$. Taking the transpose:

$S^T = (A + B + C)^T = A^T + B^T + C^T$ …

Why This Works — The Core Idea

A matrix is symmetric when it equals its own transpose. The transpose operation is linear — it distributes over addition. So if you take the transpose of a sum, you get the sum of the transposes. If every matrix in that sum is symmetric (each equals its own transpose), then the sum's transpose equals the sum itself. That's the entire argument in one line.

The question tests whether you understand this property, not whether you can compute anything. No numbers, no dimensions — just the definition and one algebraic step.

Step-by-Step Reasoning

1. Recall the definition of a symmetric matrix.

A square matrix $A$ is symmetric if $A^T = A$. This means the matrix is identical to its mirror image across the main diagonal.

2. State what we are given.

We have three matrices — call them $A$, $B$, and $C$ — all of the same order (say $n \times n$), and each is symmetric:

3. Consider the sum $S = A + B + C$.

We want to check whether $S$ is symmetric. That means we need to see if $S^T = S$.

4. Use the property of transpose under addition.

For any matrices of the same order, the transpose of a sum is the sum of the transposes:

This is a fundamental rule — it works because transposing each entry individually is a linear operation.

5. Substitute the symmetric condition. …

Method: Proving Closure Under Addition Using the Transpose Rule

Use this to decide whether a class of matrices defined by a transpose condition (symmetric, skew-symmetric) is preserved under addition. The tool is the linearity of the transpose.

Steps

Step 1: Encode the class as a transpose equation.

A matrix is symmetric when $A'=A$; skew-symmetric when $A'=-A$. Write each given matrix's defining equation.

Step 2: Apply transpose to the combination, using its distributive law.

Transpose distributes over sums: $(A+B+C)'=A'+B'+C'$. Take the transpose of the whole sum you are testing. …

Common Mistakes

Mistake 1: Assuming the product of symmetric matrices is also symmetric.

Why it's wrong: $(AB)'=B'A'=BA$, which equals $AB$ only when $A,B$ commute. Correct approach: closure holds for sums of symmetric matrices, not products in general. …