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: Concept Review:
This question assesses foundational determinant properties. We must evaluate each statement to identify the invalid one.
Step 3: Statement Analysis:
Let's examine each assertion:
(A) This is a standard determinant property. Identical rows or columns imply linear dependence, resulting in a zero determinant. This statement is valid.
(B) This is also a valid property. A row or column composed entirely of zeros yields a determinant of zero. This is evident from cofactor expansion along that row/column, as each term will include a zero factor. This statement is valid.
(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 valid.
(D) This statement is invalid. A fundamental determinant property dictates that interchanging any two rows or columns negates the determinant's sign (multiplies it by -1). The statement's assertion that the sign "remains unchanged" is incorrect.
Step 4: Conclusion:
Statement (D) is incorrect because interchanging two rows or columns reverses the determinant's sign.