Class 12 Determinants · Mcq Single Correct1 Marksmatrix_inverse
Class 12 Maths · Determinants · Mcq Single Correct #76
A real square matrix satisfies A³−5A²+4A+λI=O. Which condition guarantees invertibility for every such A?
✅ Step-by-Step Solution
Final Answer: Option (D): λ≠0
- If λ≠0, the polynomial identity gives A·[−(A²−5A+4I)/λ]=I and the reverse product also equals I, proving invertibility. If λ=0, A=O is an allowed singular example. Thus λ≠0 is exactly the condition guaranteeing invertibility for every solution A.
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