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: Conceptual Understanding:
This question assesses fundamental determinant properties. Each statement must be evaluated to identify the invalid one.

Step 3: Detailed Analysis:
Examining each statement:
(A) This is a standard determinant property. If two rows or columns are identical, they are linearly dependent, resulting in a determinant of zero. This statement is correct.

(B) This is also a correct property. A determinant with a row or column of all zeros evaluates to zero. Expansion along such a row or column confirms this, as each term would include a zero factor. This statement is correct.

(C) Swapping rows and columns is equivalent to matrix transposition. A core determinant property states that a matrix's determinant equals its transpose's determinant, i.e., \(|A| = |A^T|\). This statement is correct.

(D) This statement is incorrect. A crucial determinant property is that interchanging any two rows or columns negates the determinant (multiplies it by -1). The statement's claim that the sign "remains unchanged" is false.

Step 4: Conclusion:
Statement (D) is incorrect because interchanging two rows or columns reverses the determinant's sign.