If the determinant of a 3^{text{rd}} order ma | vedclass

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(C) Given that $A$ is a $3 	imes 3$ matrix and $|A| = K$. First,consider the matrix $(A - A^T)$. Since $(A - A^T)^T = A^T - (A^T)^T = A^T - A = -(A -

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$K^2$

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(C) Given that $A$ is a $3 	imes 3$ matrix and $|A| = K$. First,consider the matrix $(A - A^T)$. Since $(A - A^T)^T = A^T - (A^T)^T = A^T - A = -(A - A^T)$,the matrix $(A - A^T)$ is a skew-symmetric matrix. For any skew-symmetric matrix of odd order $n$,the determinant is $0$. Since $n = 3$ is odd,$|A - A^T| = 0$. Next,consider the matrix $(AA^T)$. Using the property of determinants $|XY| = |X||Y|$ and $|A^T| = |A|$,we have $|AA^T| = |A||A^T| = |A||A| = K \cdot K = K^2$. Therefore,the sum of the determinants is $|AA^T| + |A - A^T| = K^2 + 0 = K^2$.

If the determinant of a $3^{\text{rd}}$ order matrix $A$ is $K$,then the sum of the determinants of the matrices $(AA^T)$ and $(A-A^T)$ is

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