Let A be any skew- symmetric matrix (where $A^T$ is Transpose of matrix A). Then which of the following statements are correct?

(A) $A^2$ is a symmetric matrix

(B) $A^2$ is a skew- symmetric matrix

(C) $A^T A = -A^2$

(D) $A^T A - AA^T = O$

Choose the correct answer from the options given below:

A skew-symmetric matrix satisfies $A^T = -A$.

The transpose rule for matrix products: $(AB)^T = B^T A^T$ (order reverses).

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Checking Statement (A): $A^2$ is a symmetric matrix

To determine if $A^2$ is symmetric, find $(A^2)^T$:

$(A^2)^T = (A \cdot A)^T$

$= A^T \cdot A^T$

$= (-A) \cdot (-A)$

$= A^2$

Since $(A^2)^T = A^2$, the matrix $A^2$ is symmetric.

Statement (A) is correct.

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Checking Statement (B): $A^2$ is a skew-symmetric matrix

From the previous result, $(A^2)^T = A^2$.

For a matrix to be skew-symmetric, $(A^2)^T = -A^2$ must hold.

Since $(A^2)^T = A^2 \neq -A^2$, the matrix $A^2$ is not skew-symmetric.

Statement (B) is incorrect.

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Checking Statement (C): $A^T A = -A^2$

Using the skew-symmetric property $A^T = -A$:

$A^T A = (-A) \cdot A$

$= -AA$

$= -A^2$

Statement (C) is correct.

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Checking Statement (D): $A^T A - AA^T = O$

From Statement (C): $A^T A = -A^2$

Finding $AA^T$:

$AA^T = A \cdot (-A)$

$= -A^2$

Computing the difference:

$A^T A - AA^T = -A^2 - (-A^2)$

$= -A^2 + A^2$

$= O$

Statement (D) is correct.

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The correct statements are (A), (C), and (D).