Animated Solution for Mathematics - Matrices and Determinants: Let $A$ and $B$ be any two $3\times 3$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Welcome, fellow traveler, to the elegant world of matrix algebra. Today, we are exploring the fundamental nature of symmetry.

A symmetric matrix is invariant under reflection, while a skew-symmetric matrix flips its sign. Our mission is to identify the statement that is NOT true among four options involving a symmetric matrix $A$ and a skew-symmetric matrix $B$.

Before we dive into the options, let us sharpen our tools. We are given the definitions:

We also rely on the fundamental properties of the transpose operator: 1. Addition property: $(X\pm Y{)}^T=X^T\pm Y^T$ 2. Reversal law: $(XY{)}^T=Y^T{X}^T$ 3. Power property: $(X^n{)}^T=(X^T{)}^n$

Let us test our options one by one. For Option A, let $M=A^4-B^4$. Taking the transpose:

Substituting our definitions, we get:

Since $M^T=M$, this matrix is symmetric. Thus, Option A is true.

Now, for Option B, let $M=AB-BA$. Taking the transpose:

Substituting $B^T=-B$ and $A^T=A$:

Since $M^T=M$, this matrix is symmetric. Thus, Option B is true.

For Option D, let $M=AB+BA$. Taking the transpose:

Substituting the definitions:

Since $M^T=-M$, this matrix is skew-symmetric. Thus, Option D is true.

Finally, we arrive at Option C: $M=B^5-A^5$. Taking the transpose:

Substituting $B^T=-B$ and $A^T=A$:

Because the power $5$ is odd, $(-B{)}^5=-B^5$. Thus:

For this to be skew-symmetric, we would require $M^T=-M=-(B^5-A^5)=-B^5+A^5$. Clearly, $-(B^5 + A^5) eq -B^5 + A^5$.

Therefore, $B^5-A^5$ is NOT skew-symmetric, making this the false statement.