Question

Which of the following is correct?

• A. Determinant is a square matrix.
• B. Determinant is a number associated to a matrix.
• C. Determinant is a unique number associated to a square matrix.
• D. Determinant is not defined for a square matrix.

Hint:

When answering definition-based questions, look for the option that includes all necessary restrictive conditions. Option (2) is a common trap because it sounds plausible but misses the essential "square" requirement.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question tests the formal mathematical definition of a determinant, distinguishing it from the matrix itself and specifying the conditions under which it exists.

Step 2: Key Formula or Approach:
Recall the definition: A determinant is a scalar value (a real or complex number) that can be computed from the elements of a square matrix and encodes certain properties of that matrix.

Step 3: Detailed Explanation:
Let's evaluate the validity of each option:
- (1) "Determinant is a square matrix." - This statement is incorrect. A matrix is an array or table of numbers. A determinant is a single scalar value calculated from that array. They are fundamentally different types of mathematical entities.
- (2) "Determinant is a number associated to a matrix." - While a determinant is indeed a number, this statement is too broad and technically incorrect because it implies a determinant can be associated with any matrix. Determinants are not defined for rectangular matrices (where rows $\neq$ columns).
- (3) "Determinant is a unique number associated to a square matrix." - This is the precise and correct definition. The determinant operation maps every square matrix to exactly one scalar value. The requirement that the matrix be square is crucial.
- (4) "Determinant is not defined for a square matrix." - This is entirely incorrect. Square matrices are the only type of matrices for which determinants are defined.

Step 4: Final Answer:
The correct statement is option (3).