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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

X^2 = I

other than

I

is:

0

1

2

Dmore than

2

correct

Solution

Step 1: Set up the system for a

2\times 2

matrix

X

Let

X = \begin{pmatrix}a & b\\ c & d\end{pmatrix}

. Then

X^2 = I

gives:

a^2+bc = 1, \quad b(a+d) = 0, \quad c(a+d) = 0, \quad d^2+bc = 1

Step 2: Analyse Case

a = -d

The conditions

b(a+d)=0

and

c(a+d)=0

are satisfied for all

b, c

. The remaining condition is

a^2+bc = 1

. Setting

a = 0

(hence

d = 0

) requires

bc = 1

, which is satisfied by any

b \neq 0

with

c = \frac{1}{b}

. For example:

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

2

solutions other than

I

. Answer: (4)

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