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Matrices & Determinants: Number Solutions Matrix Equation Other

JEE Maths question with a full step-by-step solution.

Question
The number of solutions of the matrix equation X2=IX^2 = I other than II is:
A00
B11
C22
Dmore than 22correct
Solution
Step 1: Set up the system for a 2×22\times 2 matrix XX Let X=(abcd)X = \begin{pmatrix}a & b\\ c & d\end{pmatrix}. Then X2=IX^2 = I gives:
a2+bc=1,b(a+d)=0,c(a+d)=0,d2+bc=1a^2+bc = 1, \quad b(a+d) = 0, \quad c(a+d) = 0, \quad d^2+bc = 1
 Step 2: Analyse Case a=−da = -d The conditions b(a+d)=0b(a+d)=0 and c(a+d)=0c(a+d)=0 are satisfied for all b,cb, c. The remaining condition is a2+bc=1a^2+bc = 1. Setting a=0a = 0 (hence d=0d = 0) requires bc=1bc = 1, which is satisfied by any b≠0b \neq 0 with c=1bc = \frac{1}{b}. For example:
X=(0110),X=(02120),X=−I,…X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}, \quad X = \begin{pmatrix}0 & 2\\ \tfrac{1}{2} & 0\end{pmatrix}, \quad X = -I, \ldots
 This yields infinitely many solutions, hence more than 22 solutions other than II. Answer: (4)
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