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Matrices & Determinants — JEE Maths practice question

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

Question
If MM is a square matrix of order 2, then (tr(M))2−tr(M2)(\text{tr}(M))^2 - \text{tr}(M^2) is equal to:
Adet⁡(M)2\dfrac{\det(M)}{2}
B2det⁡(M)2\det(M)correct
C3det⁡(M)3\det(M)
D4det⁡(M)4\det(M)
Solution
Step 1: Let M=(abcd)M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
(tr(M))2=(a+d)2=a2+2ad+d2(\text{tr}(M))^2 = (a+d)^2 = a^2 + 2ad + d^2
tr(M2)=(a2+bc)+(d2+bc)=a2+d2+2bc\text{tr}(M^2) = (a^2 + bc) + (d^2 + bc) = a^2 + d^2 + 2bc
 Step 2: Subtract
(tr(M))2−tr(M2)=2ad−2bc=2(ad−bc)=2det⁡(M)(\text{tr}(M))^2 - \text{tr}(M^2) = 2ad - 2bc = 2(ad - bc) = 2\det(M)
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