Question

If \(\lambda=0\), is an eigen value of \(A\), then \(\det(A)\) is :

• A. \(0\)
• B. \(1\)
• C. \(\lambda\)
• D. Infinite

Hint:

The determinant of a matrix equals the product of its eigenvalues: \[ \det(A)=\prod \lambda_i \] If even one eigenvalue is zero, then: \[ \det(A)=0 \] and the matrix becomes singular.

Answer 1

The Correct Option is A

Concept: For a square matrix \(A\), eigenvalues and determinant are closely related. The determinant of a matrix is equal to the product of all its eigenvalues. That is: \[ \det(A)=\lambda_1\lambda_2\lambda_3\cdots\lambda_n \] Hence:
• if any eigenvalue becomes zero,
• then the entire product becomes zero. Therefore determinant immediately becomes zero.

Step 1: Using the relation between determinant and eigenvalues. Suppose the eigenvalues of matrix \(A\) are: \[ \lambda_1,\lambda_2,\lambda_3,\ldots,\lambda_n \] Then: \[ \det(A)=\lambda_1\lambda_2\lambda_3\cdots\lambda_n \]

Step 2: Using the given information. The question states: \[ \lambda=0 \] is an eigenvalue of \(A\). Therefore one factor in the product becomes zero.

Step 3: Evaluating the determinant. Since multiplication by zero gives zero: \[ \det(A)=0 \] Hence: \[ \boxed{\det(A)=0} \]

Step 4: Checking each option carefully.
• Option \(1\): \(0\) \(\rightarrow\) Correct
• Option \(2\): \(1\) \(\rightarrow\) Incorrect
• Option \(3\): \(\lambda\) \(\rightarrow\) Incorrect
• Option \(4\): Infinite \(\rightarrow\) Incorrect Therefore the correct answer is: \[ \boxed{(1)\;0} \]