Question

Which of the following statements is false?

• A. If the entries in a row of a determinant are zero, the value of determinant is zero
• B. The value of a determinant changes if the rows and columns of determinant are interchanged
• C. If any two rows of a determinant are interchanged, the value of the determinant is multiplied by -1
• D. If corresponding entries in 2 rows of a determinant are proportional, the value of the determinant is zero

Hint:

Exam Tip:
Properties of determinants:

• \(\det(A) = \det(A^T)\).
• Interchanging two rows changes sign.
• Multiplying a row by a scalar multiplies the determinant by that scalar.
• If two rows are proportional, \(\det = 0\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question tests the properties of determinants. We need to identify the false statement.

Step 2: Analyzing Each Option:

• (A) If the entries in a row of a determinant are zero, the value of determinant is zero:
This is true. Expanding along that row gives all terms multiplied by zero.
• (B) The value of a determinant changes if the rows and columns of determinant are interchanged:
This is false. The determinant of a matrix is equal to the determinant of its transpose.
So, interchanging rows and columns (transposing) does not change the value.
• (C) If any two rows of a determinant are interchanged, the value of the determinant is multiplied by -1:
This is true. This is a standard property of determinants.
• (D) If corresponding entries in 2 rows of a determinant are proportional, the value of the determinant is zero:
This is true. If two rows are proportional, one row is a scalar multiple of another, making the determinant zero.

Step 3: Final Answer:
The false statement is option (B). Therefore, option (B) is correct.