Question
If $\lambda = 0$, is an eigen values of A, then $\det(A)$ is :
This question was previously asked in
CUET PG 2026 Agri-Business Management Question Paper (25-Mar-2026) (Shift 2)
The correct answer is
0
Eigenvalue and Determinant Relationship
An eigenvalue ($\lambda$) of a square matrix $A$ is a scalar that satisfies the characteristic equation: $ \det(A - \lambda I) = 0 $ where $I$ is the identity matrix of the same dimension as $A$.
Applying the Given Eigenvalue
We are given that $\lambda = 0$ is an eigenvalue of matrix $A$. Substituting $\lambda = 0$ into the characteristic equation gives:
$ \det(A - 0 \cdot I) = 0 $Deriving the Determinant
Simplifying the expression inside the determinant:
$ A - 0 \cdot I = A $Therefore, the characteristic equation becomes:
$ \det(A) = 0 $This shows that if 0 is an eigenvalue of a matrix $A$, its determinant must be 0.
Conclusion
Hence, if $\lambda = 0$ is an eigenvalue of $A$, then $\det(A)$ is 0.
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