Verified Solution • 2025–26 Session moderate

QJEEM10Ma91d0: JEE Main 2010 Mathematics - Class 12 JEEM10Ma91d0 Solution: Let be a matrix with non-zero entries and let where is identity …

Let AA be a  2×2\,2 \times 2 matrix with non-zero entries and let A2=I,{A^2} = I,
where II is 2×22 \times 2 identity matrix. Define
TrTr (A)=(A)= sum of diagonal elements of AA and ∣A∣=\left| A \right| = determinant of matrix AA .
Statement- 1: TrTr (A)=0(A)=0 .
Statement- 2: ∣A∣=1\left| A \right| = 1 .

Multiple Choice Options

  • A statement - 1 is true, statement - 2 is true; statement - 2 is not a correct explanation for statement - 1.
  • B statement - 1 is true, statement - 2 is false.
  • C statement - 1 is false, statement -2 is true
  • D statement -1 is true, statement - 2 is true; statement - 2 is a correct explanation for statement - 1.

Step-by-Step Verified Solution

Let A=(abcd) where a,b,c,da,b,c,d ≠0\ne 0

A2=(abcd)(abcd)

⇒A2=(a2+bcab+bdac+cdbc+d2)

⇒a2+bc=1, bc+d2=1\Rightarrow {a^2} + bc = 1,\,bc + {d^2} = 1

ab+bd=ac+cd=0ab + bd = ac + cd = 0

c≠0     b≠0c \ne 0\,\,\,\,\,b \ne 0

⇒a+d=0⇒Tr(A)=0\Rightarrow a + d = 0 \Rightarrow Tr\left( A \right) = 0

∣A∣=ad−bc=−a2−bc=−1\left| A \right| = ad - bc = - {a^2} - bc = - 1

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