GATE 2023 ST – Question 12
Matrix Theory · Eigenvalues and Matrix Decompositions · 1 mark · Multiple choice
A real 3×3 matrix A has eigenvalues 1,0,−1. For B=A²+2A+I₃, which identity holds?
- B³−5B²+4B=0
- B³−5B²−4B=0
- B³+5B²−4B=0
- B³+5B²+4B=0
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Show answer and explanation
Correct answer: (A) B³−5B²+4B=0
Explanation
The eigenvalues of B are (1+1)²=4,(0+1)²=1,(−1+1)²=0. Its characteristic polynomial is t(t−1)(t−4)=t³−5t²+4t. Cayley–Hamilton yields **A**.
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