Q.If $A$ is a symmetric matrix, then $A^3$ is a _________ matrix.

The key idea is that the product of symmetric matrices is symmetric if and only if they commute. However, a symmetric matrix always commutes with itself.

Step 1: A matrix $A$ is symmetric if $A^T = A$.

Step 2: Consider $(A^3)^T = (A \cdot A \cdot A)^T$. Using the reversal property of transpose, $(A^3)^T = A^T A^T A^T$. …

The key here is to understand what "symmetric" means in matrix terms, and then see how that property behaves under multiplication.

A matrix $A$ is called symmetric if it equals its own transpose: $A^T = A$. Geometrically, this means the matrix is "mirrored" across its main diagonal — the $(i,j)$ entry equals the $(j,i)$ entry for all $i, j$.

Now, the question asks about $A^3$. The natural instinct is to check: does the symmetry survive when we multiply the matrix by itself repeatedly? The answer is yes, and the reason lies in a simple property of transposes.

This is the "reverse order" rule for transposes. It's the only tool we need.

Let's work through it step by step.

1. Start with what we know.

   We are given that $A$ is symmetric, so $A^T = A$.

2. Consider $A^3$.

   By definition, $A^3 = A \cdot A \cdot A$ (matrix multiplication).

3. Take the transpose of $A^3$.

   Using the reverse-order rule repeatedly:

The order reverses: the last $A$ becomes first, and so on.

4. Substitute the symmetry condition. Since $A^T = A$, we replace each $A^T$ with $A$:

5. Interpret the result. …

Method: Find the symmetry type of a power of a matrix

Use this to determine whether $A^n$ is symmetric when $A$ has a known symmetry.

Steps

Step 1: State the given symmetry.

Symmetric means $A' = A$.

Step 2: Transpose the power using the reversal law.

Repeated use of $(XY)' = Y'X'$ gives …

Common Mistakes

Mistake 1: Applying the transpose to a product without reversing the order.

Why it's wrong: $(A\cdot A\cdot A)' = A'A'A'$ by the reversal law; skipping this step is a habit that fails for non-identical factors. Correct approach: reverse the order every time, then use $A'=A$.

Mistake 2: Thinking an odd power might break symmetry. …