GATE 2022 ST – Question 11
A real 2×2 matrix M satisfies (I+M)⁻¹=I−αM with α≠0 and tr M=3. Find α.
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Correct answer: (D) 1/4
Explanation
Multiplication gives (1−α)M−αM²=0. If M is singular,Cayley–Hamilton gives M²=3M;since M≠0,this yields 1−4α=0 and **α=1/4, D**,the official answer. However singularity is not stated. If M is invertible,the same identity forces M=((1−α)/α)I,and trace 3 gives M=(3/2)I,α=2/5,which also satisfies the printed equation. The stated assumptions therefore do not uniquely determine the official value.