Statement (A) "If each element in a row is a constant multiplier of the corresponding element of another row of a determinant, then the value of the determinant is always non-zero." - This statement is false because if any row (or column) of a determinant is a constant multiple of another row (or column), the determinant is zero. - Mathematically, if two rows (or columns) are linearly dependent, then: \[ |A| = 0 \] Statement (B) "If each element on one side of the principal diagonal of a determinant is zero, then the value of the determinant is the product of the diagonal elements." - This describes a triangular matrix (upper or lower triangular matrix). - The determinant of a triangular matrix is given by: \[ |A| = \prod_{i=1}^{n} a_{ii} \] where \( a_{ii} \) are the diagonal elements. - Since the given condition matches this form, this statement is true. Statement (C) *"The value of determinant of a skew-symmetric matrix of odd order is always non-zero."* - A skew-symmetric matrix satisfies the property: \[ A^T = -A \] - If the order \( n \) of a skew-symmetric matrix is odd, then its determinant is always zero, i.e., \[ \det(A) = 0 \] - Since the statement claims it is non-zero, it is false. Statement (D) *"If \( A \) is a non-singular matrix of order three, then \( |\text{adj} A| = |A|^2 \)."* - For any non-singular square matrix \( A \) of order \( n \), the determinant of the adjugate is given by: \[ |\text{adj}(A)| = |A|^{n-1} \] - For a 3×3 matrix: \[ |\text{adj}(A)| = |A|^{3-1} = |A|^2 \] - Since the given statement correctly follows this property, it is true. Final Answer: - The correct statements are (B) and (D). - Thus, the correct answer is: \[ \boxed{\text{(B) and (D) only}} \]