Question

Which of the following statements is incorrect?

• A. If two rows or two columns of a determinant are identical, then the value of the determinant is zero.
• B. If all the elements in any one row of the determinant are zero, then the determinant value is zero.
• C. The value of the determinant remains unchanged if its rows and columns are interchanged.
• D. If any two rows of a determinant are interchanged, then the sign of the determinant remains unchanged.

Hint:

The properties of determinants are crucial for both computational problems and theoretical questions. It is highly recommended to memorize the 6-7 key properties, especially those related to row/column operations.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests the fundamental properties of determinants. We need to evaluate each statement to identify the one that is not a valid property.
Step 3: Detailed Explanation:
Let's analyze each statement:
(A) This is a standard property of determinants. If two rows or two columns of a matrix are identical, the rows/columns are linearly dependent, and the determinant is zero. This statement is correct.
(B) This is also a correct property. If any row or column of a determinant contains all zeros, its value is zero. This can be easily seen by expanding the determinant along that row or column. Each term in the expansion would have a zero factor. This statement is correct.
(C) Interchanging the rows and columns of a matrix is equivalent to taking its transpose. A fundamental property of determinants is that the determinant of a matrix is equal to the determinant of its transpose, i.e., \(|A| = |A^T|\). This statement is correct.
(D) This statement is incorrect. A key property of determinants is that if any two rows or any two columns are interchanged, the sign of the determinant is changed (it is multiplied by -1). The statement claims the sign "remains unchanged," which is false.
Step 4: Final Answer:
The incorrect statement is (D) because interchanging two rows or columns reverses the sign of the determinant.